zero and negative exponents common core algebra 1 homework answers

Algebra 1: Common Core (15th Edition)

By charles, randall i., chapter 7 - exponents and exponential functions - 7-1 zero and negative exponents - lesson check - page 421: 1, work step by step, update this answer.

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Algebra 1 Common Core

Randall i. charles,basia hall, dan kennedy, exponents and exponential functions - all with video answers.

Zero and Negative Exponents

Simplify each expression. $$ 2^{-5} $$

Simplify each expression. $$ m^{0} $$

Simplify each expression. $$ 5 s^{2} t^{-1} $$

Simplify each expression. $$ \frac{4}{x^{-3}} $$

Evaluate each expression for $a=2$ and $b=-4$ $$ \text { Evaluate each expression for } a=2 \text { and } b=-4 $$

Evaluate each expression for $a=2$ and $b=-4$ $$ 2 a^{-4} b^{0} $$

Vocabulary A positive exponent shows repeated multiplication. What repeated operation does a negative exponent show?

Error Analysis A student incorrectly simplified $\frac{x^{n}}{a-n n^{0} r}$ as shown below. Find and correct the student's error.

Simplify each expression. $$ 3^{-2} $$

Simplify each expression. $$ (-4.25)^{0} $$

Simplify each expression. $$ (-5)^{-2} $$

Simplify each expression. $$ -5^{-2} $$

Simplify each expression. $$ (-4)^{-2} $$

Simplify each expression. $$ 2^{-6} $$

Simplify each expression. $$ -3^{0} $$

Simplify each expression. $$ -12^{-1} $$

Simplify each expression. $$ \frac{1}{2^{0}} $$

Simplify each expression. $$ 58^{-1} $$

Simplify each expression. $$ 1.5^{-2} $$

Simplify each expression. $$ (-5)^{-3} $$

Simplify each expression. $$ 4 a b^{0} $$

Simplify each expression. $$ \frac{1}{x^{-7}} $$

Simplify each expression. $$ 5 x^{-4} $$

Simplify each expression. $$ \frac{1}{c^{-1}} $$

Simplify each expression. $$ \frac{3^{-2}}{n} $$

Simplify each expression. $$ k^{-4} j^{0} $$

Simplify each expression. $$ \frac{3 x^{-2}}{y} $$

Simplify each expression. $$ \frac{7 a b^{-2}}{3 w} $$

Simplify each expression. $$ c^{-5} d^{-7} $$

Simplify each expression. $$ c^{-5} d^{7} $$

Simplify each expression. $$ \frac{8}{2 s^{-3}} $$

Simplify each expression. $$ \frac{7 s}{5 t^{-3}} $$

Simplify each expression. $$ \frac{6 a^{-1} c^{-3}}{d^{0}} $$

Simplify each expression. $$ 2^{-3} x^{2} z^{-7} $$

Simplify each expression. $$ 12^{0} t^{7} u^{-11} $$

Simplify each expression. $$ \frac{7 s^{0} t^{-5}}{2^{-1} m^{2}} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ r^{-3} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ s^{-3} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ \frac{3 r}{s^{-2}} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ \frac{s^{0}}{r^{-2}} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ 4 s^{-1} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ r^{0} s^{-2} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ r^{-4} s^{2} $$

Evaluate each expression for $r=-3$ and $s=5$ $$ 2^{-4} r^{3} s^{-2} $$

Internet Traffic The number of visitors to a certain web site triples every month. The number of visitors is modeled by the expression $8100 \cdot 3^{m}$, where $m$ is the number of months after the number of visitors was measured. Evaluate the expression $m=-4 .$ What does the value of the expression represent in the situation?

Population Growth A Galápagos cactus finch population increases by half every decade. The number of finches is modeled by the expression $45 \cdot 1.5^{d},$ where $d$ is the number of decades after the population was measured. Evaluate the expression for $d=-2, d=0$, and $d=1 .$ What does each value of the expression represent in the situation?

Mental Math Is the value of each expression positive or negative? $$ -2^{2} $$

Mental Math Is the value of each expression positive or negative? $$ (-2)^{2} $$

Mental Math Is the value of each expression positive or negative? $$ (-2)^{3} $$

Mental Math Is the value of each expression positive or negative? $$ (-2)^{-3} $$

Write each number as a power of 10 using negative exponents. $$ \frac{1}{10} $$

Write each number as a power of 10 using negative exponents. $$ \frac{1}{100} $$

Write each number as a power of 10 using negative exponents. $$ \frac{1}{1000} $$

Write each number as a power of 10 using negative exponents. $$ \frac{1}{10,000} $$

a. Patterns Complete the pattern using powers of 5. $$ \frac{1}{5^{2}}=\mathbb{} \quad \frac{1}{5^{1}}=\mathbb{} \quad \frac{1}{5^{0}}=\mathbb{} \quad \frac{1}{5^{-1}}=\mathbb{} \quad \frac{1}{5^{2}}=\mathbb{} $$ b. Write $\frac{1}{5^{4}}$ using a positive exponent. c. Rewrite $\frac{1}{a^{-n}}$ as a power of $a$.

Rewrite each fraction with all the variables in the numerator. $$ \frac{a}{b^{-2}} $$

Rewrite each fraction with all the variables in the numerator. $$ \frac{4 g}{h^{3}} $$

Rewrite each fraction with all the variables in the numerator. $$ \frac{5 m^{6}}{3 n} $$

Rewrite each fraction with all the variables in the numerator. $$ \frac{8 c^{5}}{11 d^{4} e^{-2}} $$

Think About a Plan Suppose your drama club's budget doubles every year. This year the budget is 500 .dollar How much was the club's budget 2 yr ago? $\cdot$ What epression models what the budget of the club will be in $1 \mathrm{yr} ?$ In $2 \mathrm{yr}$ ? In years? $\cdot$ What value fy can you substitute into your expression to find the budget of the club $2 \mathrm{yr}$ ago?

Copy and complete the table at the right. (Table cannot copy)

a. Simplify $a^{n} \cdot a^{-n}$. b. Reasoning what is the mathematical relationship between $a^{n}$ and $a^{-n}$ ? Explain.

Open-Ended Choose a fraction to use as a value for the variable $a$. Find the values of $a^{-1}, a^{2},$ and $a^{-2}$.

Manufacturing A company is making metal rods with a target diameter of 1.5 $\mathrm{mm}$. A rod is acceptable when its diameter is within $10^{-3} \mathrm{mm}$ of the target diameter. Write an inequality for the acceptable range of diameters.

Reasoning Are $3 x^{-2}$ and $3 x^{2}$ reciprocals? Explain.

Simplify each expression. $$ \left(\frac{r^{-7} b^{-8}}{t^{-4} u^{1}}\right)^{0} $$

Simplify each expression. $$ (-5)^{2}-(0.5)^{-2} $$

Simplify each expression. $$ \frac{6}{m^{2}}+\frac{5 m^{-2}}{3^{-3}} $$

Simplify each expression. $$ 2^{3}\left(5^{0}-6 m^{2}\right) $$

Simplify each expression. $$ \frac{2 x^{-5} y^{3}}{n^{2}} \div \frac{r^{2} y^{5}}{2 n} $$

Simplify each expression. $$ 2^{-1}-\frac{1}{3^{-2}}+5\left(\frac{1}{2^{2}}\right) $$

For what value or values of $n$ is $n^{-3}=\left(\frac{1}{n}\right)^{5} ?$

What is the simplified form of $-6(-6)^{-1} ?$

Segment CD represents the flight of a bird that passes through the points $(1,2)$ and $(5,4)$. What is the slope of a line that represents the flight of a second bird that flew perpendicular to the first bird?

What is the solution of the equation $1.5(x-2.5)=3 ?$

What is the simplified form of $|3.5-4.7|+5.6 ?$

What is the $y$ -intercept of the graph of $3 x-2 y=-8 ?$

Solve each system by graphing. $$ \begin{array}{l}{y>3 x+4} \\ {y \leq-3 x+1}\end{array} $$

Solve each system by graphing. $$ \begin{array}{l}{y \leq-2 x+1} \\ {y<2 x-1}\end{array} $$

Solve each system by graphing. $$ \begin{array}{l}{y \geq 0.5 x} \\ {y \leq x+2}\end{array} $$

Write an equation in slope-intercept form for the line with the given slope $m$ and $y$ -intercept $b$. $$ m=-1, b=4 $$

Write an equation in slope-intercept form for the line with the given slope $m$ and $y$ -intercept $b$. $$ m=5, b=-2 $$

Write an equation in slope-intercept form for the line with the given slope $m$ and $y$ -intercept $b$. $$ m=\frac{2}{5}, b=-3 $$

Write an equation in slope-intercept form for the line with the given slope $m$ and $y$ -intercept $b$. $$ m=-\frac{3}{11}, b=-17 $$

Write an equation in slope-intercept form for the line with the given slope $m$ and $y$ -intercept $b$. $$ m=\frac{5}{9}, b=\frac{1}{3} $$

Write an equation in slope-intercept form for the line with the given slope $m$ and $y$ -intercept $b$. $$ m=1.25, b=-3.79 $$

To prepare for Lesson $7-2,$ do Exercises $87-91$. Simplify each expression. $$ 6 \cdot 10^{4} $$

Simplify each expression. $$ 7 \cdot 10^{-2} $$

Simplify each expression. $$ 8.2 \cdot 10^{5} $$

Simplify each expression. $$ 3 \cdot 10^{-3} $$

Simplify each expression. $$ 3.4 \cdot 10^{5} $$

Common Core Algebra I Math (Worksheets, Homework, Lesson Plans)

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Negative Exponents and Zero Exponents

So far in this unit, you've learned how to simplify monomial expressions with positive exponents. Now we are going to study two more aspects of monomials: those that have negative exponents and those that have zero as an exponent .

I am going to let you investigate to see if you can come up with the rule on your own! Take a look at the following problems and see if you can determine the pattern.

Can you figure out the rule? If not, here it is...

The Rule for Negative Exponents:

The expression a -n is the reciprocal of a n

A reciprocal is when you "flip a fraction".

The reciprocal of 3/4 is 4/3.

The reciprocal of 5 is 1/5. (You can make a whole number a fraction by putting a one in the denominator: 5 = 5/1)

***An easy rule to remember is: if the number is in the numerator (top), move it to the denominator (bottom). If the number is in the denominator, move it to the numerator!

Let's take a look at a couple of examples:

Examples of Negative Exponents

Now let's quickly take a look at monomials that contain the exponent 0.

Any number (except 0) to the zero power is equal to 1.

Not too hard, is it? Let's look at a couple of example problems and then you can practice a few.

Example 1: Negative Exponents

Example 2: evaluating negative exponents.

**Since 2/3 is in parenthesis, we must apply the power of a quotient property and raise both the 2 and 3 to the negative 2 power.

First take the reciprocal to get rid of the negative exponent.

Then raise (3/2) to the second power.

Now, it's going to get a little more tough.

Example 3: Complex Expressions with Negative Exponents

One more example.

Example 4: More Negative Exponents

Yes, I know that's a lot of examples to comprehend. My goal was to start easy and progress to harder problems. Are you ready to try a few on your own?

Practice Problems

So, how did you do? Are you ready to move onto Scientific Notation ?

• Negative Exponents

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Chapter 5: Exponential Functions

5.3.2: properties of exponents with zero and negative exponents, learning outcomes.

• Explain the meaning of a zero exponent
• Explain the meaning of a negative exponent
• Apply the power of product rule for exponents
• Apply the power of quotient rule for exponents
• Simplify exponential expressions
• Use properties of exponents to write equivalent exponential functions in standard form

Zero Exponents

The quotient rule for exponents can be used to determine the meaning of [latex]x^0[/latex]:

The quotient rule for exponents tells us that by subtracting the exponents:

[latex]\dfrac{x^n}{x^n}=x^0[/latex]

But since the numerator and denominator are identical, we can cancel the terms by division:

[latex]\dfrac{x^n}{x^n}=1[/latex] provided [latex]x\neq0[/latex] since we can’t divide by 0.

[latex]x^0=1[/latex], [latex]x\neq0[/latex]

exponent of zero

For all real numbers [latex]a\neq0[/latex],

[latex]a^0=1[/latex]

For example,

[latex]\begin{aligned}2022^0&=1\\(pq)^0&=1,\;p,\;q\neq 0 \\(2050xy)^{0}&=1,\;x,\;y\neq 0\end{aligned}[/latex]

[latex]\begin{aligned}\dfrac{5 a^m z^2}{a^mz}&=5\cdot\dfrac{a^m}{a^m}\cdot\dfrac{z^2}{z}&&\text{Separate into fractions}\\&=5 \cdot a^{m-m}\cdot z^{2-1}&&\text{Subtract the exponents}\\&=5\cdot a^0 \cdot z^1 &&\text{Simplify }a^0=1\text{ and }z^1=z\\&=5z\end{aligned}[/latex]

Simplify each expression.

• [latex]\dfrac{{c}^{3}}{{c}^{3}}[/latex]
• [latex]\dfrac{-3{x}^{5}}{{x}^{5}}[/latex]
• [latex]\dfrac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex]
• [latex]\dfrac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex]

We can apply the zero exponent rule and other rules to simplify each expression:

Simplify each expression using the zero exponent rule of exponents.

• [latex]\dfrac{{t}^{7}}{{t}^{7}}[/latex]
• [latex]\dfrac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}[/latex]
• [latex]\dfrac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}[/latex]
• [latex]\dfrac{-5{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}[/latex]
• [latex]1[/latex]
• [latex]\dfrac{1}{2}[/latex]
• [latex]-5[/latex]

Negative Exponents

The quotient rule for exponents can also be used to determine what it means to have a negative exponent [latex]x^{-n}[/latex]. If [latex]m<n,\;m-n<0[/latex], so [latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex] will have a negative exponent.

Consider, for example, [latex]\dfrac{{x}^{2}}{{x}^{4}}={x}^{2-4}=x^{-2}[/latex]. We have a negative exponent, but what does that mean?

Another way to simplify [latex]\dfrac{{x}^{2}}{{x}^{4}}[/latex] is to expand the numerator and denominator then cancel common factors:

[latex]\begin{align}\frac{x^{2}}{x^{4}} &=\frac{x\cdot x}{x\cdot x\cdot x\cdot x} \\[1mm] &=\frac{\cancel{x}\cdot\cancel{x}}{\cancel{x}\cdot\cancel{x}\cdot x\cdot x} \\[1mm] & =\frac{1}{x^2}\end{align}[/latex]

Consequently, [latex]x^{-2}=\dfrac{1}{x^2}[/latex].

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator. This can be generalized to [latex]a^{-n}=\dfrac{1}{a^n}[/latex].

If the negative exponent is on the denominator, [latex]\dfrac{1}{x^{-n}}[/latex], we can use division of fractions to simplify it:

[latex]\begin{aligned}\dfrac{1}{x^{-n}}&=1\div x^{-n}\\&=1\div \dfrac{1}{x^n}\\&=1\times \dfrac{x^n}{1}\\&=x^n\end{aligned}[/latex]

NEGATIVE EXPONENTS

For any real numbers [latex]a\neq0[/latex] and [latex]n[/latex],

[latex]a^{-n}=\dfrac{1}{a^n}[/latex]

[latex]\dfrac{1}{{a}^{-n}}=a^n[/latex]

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar.

Simplify the expressions. Write answers with positive exponents.

• [latex]\dfrac{x^3}{x^{10}}[/latex]
• [latex]\dfrac{z^2\cdot z}{z^4}[/latex]
• [latex]\dfrac{{\left(-5t^3\right)}^4}{\left(-5t^3\right)^8}[/latex]

• [latex]\dfrac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}[/latex]
• [latex]\dfrac{{f}^{47}}{{f}^{49}\cdot f}[/latex]
• [latex]\dfrac{2{k}^{4}}{5{k}^{7}}[/latex]
• [latex]\dfrac{\left(-3x^4\right)^5}{\left(-3x^4\right)^{12}}[/latex]
• [latex]\dfrac{5y^{-8}}{y^{-6}}[/latex]
• [latex]\dfrac{1}{{\left(-3t\right)}^{6}}[/latex]
• [latex]\dfrac{1}{{f}^{3}}[/latex]
• [latex]\dfrac{2}{5{k}^{3}}[/latex]
• [latex]\dfrac{1}{\left(-3x^4\right)^7}[/latex]
• [latex]\dfrac{5}{y^2}[/latex]

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

• [latex]{b}^{2}\cdot {b}^{-8}[/latex]
• [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex]
• [latex]\dfrac{-7z}{{\left(-7z\right)}^{5}}[/latex]

Simplify. Write answers with positive exponents.

• [latex]{t}^{-11}\cdot {t}^{6}[/latex]
• [latex]\dfrac{{25}^{12}}{{25}^{13}}[/latex]
• [latex]{t}^{-5}=\dfrac{1}{{t}^{5}}[/latex]
• [latex]\dfrac{1}{25}[/latex]

The Power of a Product Rule

To simplify the power of a product of two exponential expressions, we can use the  power of a product rule of exponents ,  which breaks up the power of a product of factors into the product of the powers of the factors. For example, consider [latex]{\left(pq\right)}^{3}[/latex]. We begin by using the associative and commutative properties of multiplication to regroup the factors:

In other words, [latex]{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}[/latex].

For any real numbers [latex]a,\;b[/latex] and [latex]n[/latex], the power of a product rule of exponents states that

Simplify each of the following products as much as possible. Write answers with positive exponents.

• [latex]{\left(a{b}^{2}\right)}^{3}[/latex]
• [latex]{\left(2t\right)}^{15}[/latex]
• [latex]{\left(-2{w}^{3}\right)}^{3}[/latex]
• [latex]\dfrac{1}{{\left(-7z\right)}^{4}}[/latex]
• [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex]

We can use the product and quotient rules and the new definitions to simplify each expression. If a number is raised to a power, we can evaluate it.

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

• [latex]{\left({g}^{2}{h}^{3}\right)}^{5}[/latex]
• [latex]{\left(5t\right)}^{3}[/latex]
• [latex]{\left(-3{y}^{5}\right)}^{3}[/latex]
• [latex]\dfrac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}[/latex]
• [latex]{\left({r}^{3}{s}^{-2}\right)}^{4}[/latex]
• [latex]{g}^{10}{h}^{15}[/latex]
• [latex]125{t}^{3}[/latex]
• [latex]-27{y}^{15}[/latex]
• [latex]\dfrac{1}{{a}^{18}{b}^{21}}[/latex]
• [latex]\dfrac{{r}^{12}}{{s}^{8}}[/latex]

The Power of a Quotient Rule

To simplify the power of a quotient of two expressions, we can use the  power of a quotient rule ,  which states that the power of a quotient of factors is the quotient of the powers of the factors.The power of a quotient rule is an extension of the power of a product rule since a quotient can be written as a product:

[latex]\begin{aligned}\left(\dfrac{a}{b}\right)^n&=\left(a\times\dfrac{1}{b}\right)^n\\&=a^n\times \left(b^{-1}\right)^n\\&=a^n\times b^{-n}\\&=a^n\times \dfrac{1}{b^n}\\&=\dfrac{a^n}{b^n}\end{aligned}[/latex].

For any real numbers [latex]a,\;b[/latex] and [latex]n[/latex], provided [latex]b\neq0[/latex], the power of a quotient rule of exponents states that

Simplify each of the following quotients as much as possible. Write answers with positive exponents.

• [latex]{\left(\dfrac{4}{{z}^{11}}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{p}{{q}^{3}}\right)}^{6}[/latex]
• [latex]{\left(\dfrac{-1}{{t}^{2}}\right)}^{27}[/latex]
• [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex]
• [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex]

• [latex]{\left(\dfrac{{b}^{5}}{c}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{5}{{u}^{8}}\right)}^{4}[/latex]
• [latex]{\left(\dfrac{-1}{{w}^{3}}\right)}^{35}[/latex]
• [latex]{\left({p}^{-4}{q}^{3}\right)}^{8}[/latex]
• [latex]{\left({c}^{-5}{d}^{-3}\right)}^{4}[/latex]
• [latex]\dfrac{{b}^{15}}{{c}^{3}}[/latex]
• [latex]\dfrac{625}{{u}^{32}}[/latex]
• [latex]\dfrac{-1}{{w}^{105}}[/latex]
• [latex]\dfrac{{q}^{24}}{{p}^{32}}[/latex]
• [latex]\dfrac{1}{{c}^{20}{d}^{12}}[/latex]

Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combining terms or exponents. Evaluating an expression means to get a numerical answer. The rules for exponents can be combined to simplify expressions.

Simplify each expression and write the answer with positive exponents only.

• [latex]{\left(6{m}^{2}{n}^{-1}\right)}^{3}[/latex]
• [latex]{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}[/latex]
• [latex]{\left(\dfrac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}[/latex]
• [latex]\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)[/latex]
• [latex]{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}[/latex]
• [latex]\dfrac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}[/latex]

• [latex]{\left(2u{v}^{-2}\right)}^{-3}[/latex]
• [latex]{x}^{8}\cdot {x}^{-12}\cdot x[/latex]
• [latex]{\left(\dfrac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}[/latex]
• [latex]\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)[/latex]
• [latex]{\left(\dfrac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\dfrac{4}{9}t{w}^{-2}\right)}^{3}[/latex]
• [latex]\dfrac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}[/latex]
• [latex]\dfrac{{v}^{6}}{8{u}^{3}}[/latex]
• [latex]\dfrac{1}{{x}^{3}}[/latex]
• [latex]\dfrac{{e}^{4}}{{f}^{4}}[/latex]
• [latex]\dfrac{27r}{s}[/latex]
• [latex]\dfrac{16{h}^{10}}{49}[/latex]

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Simplifying Exponential Functions

Each of the properties of exponents can be used to simplify and write equivalent exponential functions. For example, the function [latex]f(x)=2^{x+3}[/latex] can be written as the equivalent exponential function [latex]f(x)=8\left(2^x\right)[/latex]:

[latex]\begin{aligned}f(x)&=2^{x+3}\\&=2^x\cdot 2^3\\&=2^x\cdot 8\\&=8\left(2^x\right)\end{aligned}[/latex]

Being able to simplify an exponential function into the standard form [latex]f(x)=ar^{x-h}+k[/latex] makes the function easier to graph and allows us to determine the transformations that were made from the parent function [latex]f(x)=r^x[/latex]. In the case of [latex]f(x)=2^{x+3}=8\left(2^x\right)[/latex], the 8 tells us that the graph of [latex]f(x)=2^x[/latex] has been stretched by a factor of 8.

Use exponential rules to write equivalent exponential functions .  

• [latex]f(x)=3^{x+2}[/latex]
• [latex]g(x)=5^{2x+1}[/latex]
• [latex]h(x)=4^{3x-2}[/latex]

[latex]\begin{aligned}f(x)&=3^{x+2}\\&=3^x\cdot 3^2&&\text{Product rule (in reverse)}\\&=3^x\cdot 9&&\text{Evaluate }3^2=9\\&=9\left(3^x\right)&&\text{Write in standard form}\end{aligned}[/latex]

[latex]\begin{aligned}g(x)&=5^{2x+1}\\&=5^{2x}\cdot 5^1&&\text{Product rule (in reverse)}\\&=\left(5^2\right)^x\cdot 5&&\text{Evaluate }5^1=5\\&=5\left(25^x\right)&&\text{Write in standard form}\end{aligned}[/latex]

[latex]\begin{aligned}h(x)&=4^{3x-2}\\&=4^{3x}\cdot 4^{-2}&&\text{Product rule (in reverse)}\\&=\left(4^3\right)^x\cdot \dfrac{1}{4^2}&&\text{Power to a power rule (in reverse) and negative exponent rule}\\&=64^x\cdot \dfrac{1}{16}&&\text{Evaluate: }4^3=64\text{ and }4^2=16\\&=\dfrac{1}{16}\left(64^x\right)&&\text{Write in standard form}\end{aligned}[/latex]

• [latex]f(x)=2^{x+4}[/latex]
• [latex]g(x)=3^{x-2}[/latex]
• [latex]h(x)=5^{2x-1}[/latex]
• [latex]f(x)=16\left(2^x\right)[/latex]
• [latex]g(x)=\dfrac{1}{9}\left(3^x\right)[/latex]
• [latex]h(x)=\dfrac{1}{5}\left(25^{x}\right)[/latex]
• Adaptation and Revision. Authored by : Hazel McKenna and Leo Chang. Provided by : Utah Valley University. License : CC BY: Attribution
• Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
• Simplifying Exponential Functions. Authored by : Hazel McKenna. Provided by : Utah Valley University. License : CC BY: Attribution
• Try It: hjm292. Authored by : Hazel McKenna. Provided by : Utah Valley University. License : CC BY: Attribution
• College Algebra. Authored by : Abramson, Jay et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Simplify Expressions With Zero Exponents. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/rpoUg32utlc . License : Public Domain: No Known Copyright
• Simplify Expressions With Negative Exponents. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/Gssi4dBtAEI . License : CC BY: Attribution
• Power of a Product. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/p-2UkpJQWpo . License : CC BY: Attribution
• Power of a Quotient. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/BoBe31pRxFM . License : CC BY: Attribution
• Question ID 44120, 43231. Authored by : Brenda Gardner. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• Question ID 7833, 14060. Authored by : Tyler Wallace. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• Question ID 109762, 109765. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• Question ID 51959. Authored by : Roy Shahbazian. License : CC BY: Attribution
• Question ID 93393. Authored by : Michael Jenck. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• Question ID 14047, 14058, 14059, 14046, 14051, 14056, 14057. Authored by : James Souza. License : CC BY: Attribution
• Question ID 43896. Authored by : Carla Kulinsky. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• College Algebra. Authored by : OpenStax College Algebra. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:1/Preface . License : CC BY: Attribution

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• Algebra1 Worksheet
• Algebra 1 - Worksheet 13 - Zero and . . .

Algebra 1 - Worksheet 13 - Zero and Negative Exponents

Zero and Negative Exponents: Eighth Grade Math Lesson

• Ginean Royal
• Categories : Lesson plans for middle school math
• Tags : Teaching middle school grades 6 8

Once the students have learned the basics of exponents, (base, exponent, power, factor) and how to multiply two or more powers that have the same base, and how to raise exponents to a power when finding the quotient, now the student will work with exponents of zero and negative exponents.

Common Core State Standards

A.SSE.2: Use the structure of an expression to identify ways to rewrite it.

F.IF.8b: Use the properties of exponents to interpret expressions for exponential functions

Mathematical Practice(s): 2. Reason abstractly and quantitatively.

Learning Target(s)

• I can explain why equivalent expressions are equivalent.
• I can look for and identify clues in the structure of expressions in order to rewrite it another way.

Essential Question(s)

Why structure expressions in different ways?

Vocabulary: monomial, equivalent expressions, base, exponent, power, factor, quotient

• Review the vocabulary.
• Add to the Foldable .
• Write Zero exponents on the tab below Power of a Quotient and provide a basic example. Ex: x0 = 1, 30 = 1
• Now lift up the tab and write on the top portion – Anything to the zero power is 1. If the base is a variable, the base is gone and the answer is just 1. On the bottom portion that says, “Zero Exponent”, provide two examples and explain the examples in detail.
• (Continual reminder): if no exponent is written, the exponent is one (1).
• Write Negative Exponent on the tab below Zero Exponent and provide two basic examples. Ex1:5-2 = 1 / 52 OR 1 / 25; Ex 2: 1 / m-3 = m3 / 1 = m3
• Now lift up the tab and write on the top portion – When the exponent is negative, take the reciprocal of the term. On the bottom portion that says, “Negative Exponent”, provide two examples and explain the examples in detail.
• REMINDER: Inform students that the denominator should not equal zero.
• NOTE: As a first step when you have negative exponents, it may be easier to write the problem as multiplying fractions (with 1 in the numerator or denominator) before taking the reciprocal. EX: x-2y3 / z-4 = (x-2/1)(y3/1)(1/z -4) = y3z4 / x2

* Now you have the basis of your lesson and you can move on to Guided Practice.

Guided Practice: 3-6 practice problems. You can do 1or 2 problems with the students at the board (Smart Board, Elmo, etc.) and then put them in small groups of no more than 3 to do the rest. These problems can be pulled from any textbook or other resource.

Independent Practice: Approximately 5 problems to be done alone.

Closure/Review: Ask 1-3 questions relating to today’s lesson to be answered by the class as a whole. This will give you a general idea of the class’ understanding of today’s topic.

Exit Ticket: This is to be done the last 3-5 minutes of class and given to you (by hand or in a designated area of your room) as they leave class.

Possible question(s):

• What is your first step in solving this problem? 4-3
• What is your first step in solving this problem? 4x-3
• Can any connection be made between the two previous problems?

Below is the entire foldable with examples in a Word document. Each day you can add to the foldable and at the end of the lessons/unit, you will have notes for each area in one location. This attachment will be at the end of each lesson for Laws of Exponents.

<strong>Foldable</strong>

(Foldables are interactive organizers created by Dinah Zike). This foldable is the Layered-Look Book.

This post is part of the series: The Laws of Exponents

This series goes beyond the basics with your 8th grade students.

• Laws of Exponents Lesson One: Product of Powers
• Laws of Exponents Two: Power of a Power &amp; Power of a Product
• Laws of Exponents Three: Quotient of a Power &amp; Power of a Quotient
• Laws of Exponents Four: Zero and Negative Exponents

• $ 0.00 0 items

Unit 6 – Exponential Algebra and Functions

Exponential Increase and Decrease

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

SMART NOTEBOOK

Geometric Sequences

Equivalent Exponential Expressions

Simplifying Fractions Involving Exponents

Zero and Negative Exponents

More Work with Exponent Properties

Introduction to Exponential Functions

Percent Review

Percent Increase and Decrease

Exponential Models Based on Percent Growth

Constructing Exponential Functions

Linear Versus Exponential Functions

Unit Review

Unit 6 Review

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit 6 Assessment – Form A

EDITABLE ASSESSMENT

Unit 6 Assessment – Form B

Unit 6 Exit Tickets

Unit 6 Mid-Unit Quiz – Form A

U06.AO.01 – Investment Options Modeling Task

PDF DOCUMENT - SPANISH

EDITABLE RESOURCE

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Study Guides > College Algebra CoRequisite Course

Zero and negative exponents, learning outcomes.

• Simplify expressions with exponents equal to zero.
• Simplify expressions with negative exponents.
• Simplify exponential expressions.

[latex]\dfrac{t^{8}}{t^{8}}=\dfrac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex]

A General Note: The Zero Exponent Rule of Exponents

Using order of operations with fractions.

[latex]\dfrac{5 a^m z^2}{a^mz}\quad=\quad 5\cdot\dfrac{a^m}{a^m}\cdot\dfrac{z^2}{z} \quad=\quad 5 \cdot a^{m-m}\cdot z^{2-1}\quad=\quad 5\cdot a^0 \cdot z^1 \quad=\quad 5z[/latex]

Example: Using the Zero Exponent Rule

• [latex]\dfrac{{c}^{3}}{{c}^{3}}[/latex]
• [latex]\dfrac{-3{x}^{5}}{{x}^{5}}[/latex]
• [latex]\dfrac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex]
• [latex]\dfrac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex]
• [latex]\begin{align}\frac{c^{3}}{c^{3}} & =c^{3-3} \\ & =c^{0} \\ & =1\end{align}[/latex]
• [latex]\begin{align} \frac{-3{x}^{5}}{{x}^{5}}& = -3\cdot \frac{{x}^{5}}{{x}^{5}} \\ & = -3\cdot {x}^{5 - 5} \\ & = -3\cdot {x}^{0} \\ & = -3\cdot 1 \\ & = -3 \end{align}[/latex]
• [latex]\begin{align} \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& = \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}} && \text{Use the product rule in the denominator}. \\ & = \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}} && \text{Simplify}. \\ & = {\left({j}^{2}k\right)}^{4 - 4} && \text{Use the quotient rule}. \\ & = {\left({j}^{2}k\right)}^{0} && \text{Simplify}. \\ & = 1 \end{align}[/latex]
• [latex]\begin{align} \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& = 5{\left(r{s}^{2}\right)}^{2 - 2} && \text{Use the quotient rule}. \\ & = 5{\left(r{s}^{2}\right)}^{0} && \text{Simplify}. \\ & = 5\cdot 1 && \text{Use the zero exponent rule}. \\ & = 5 && \text{Simplify}. \end{align}[/latex]
• [latex]\dfrac{{t}^{7}}{{t}^{7}}[/latex]
• [latex]\dfrac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}[/latex]
• [latex]\dfrac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}[/latex]
• [latex]\dfrac{{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}[/latex]
• [latex]1[/latex]
• [latex]\dfrac{1}{2}[/latex]

Using the Negative Rule of Exponents

A general note: the negative rule of exponents, example: using the negative exponent rule.

• [latex]\dfrac{{\theta }^{3}}{{\theta }^{10}}[/latex]
• [latex]\dfrac{{z}^{2}\cdot z}{{z}^{4}}[/latex]
• [latex]\dfrac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}[/latex]
• [latex]\dfrac{{\theta }^{3}}{{\theta }^{10}}={\theta }^{3 - 10}={\theta }^{-7}=\dfrac{1}{{\theta }^{7}}[/latex]
• [latex]\dfrac{{z}^{2}\cdot z}{{z}^{4}}=\dfrac{{z}^{2+1}}{{z}^{4}}=\dfrac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\dfrac{1}{z}[/latex]
• [latex]\dfrac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4 - 8}={\left(-5{t}^{3}\right)}^{-4}=\dfrac{1}{{\left(-5{t}^{3}\right)}^{4}}[/latex]
• [latex]\dfrac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}[/latex]
• [latex]\dfrac{{f}^{47}}{{f}^{49}\cdot f}[/latex]
• [latex]\dfrac{2{k}^{4}}{5{k}^{7}}[/latex]
• [latex]\dfrac{1}{{\left(-3t\right)}^{6}}[/latex]
• [latex]\dfrac{1}{{f}^{3}}[/latex]
• [latex]\dfrac{2}{5{k}^{3}}[/latex]

Example: Using the Product and Quotient Rules

• [latex]{b}^{2}\cdot {b}^{-8}[/latex]
• [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex]
• [latex]\dfrac{-7z}{{\left(-7z\right)}^{5}}[/latex]
• [latex]{b}^{2}\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\frac{1}{{b}^{6}}[/latex]
• [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5 - 5}={\left(-x\right)}^{0}=1[/latex]
• [latex]\dfrac{-7z}{{\left(-7z\right)}^{5}}=\dfrac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1 - 5}={\left(-7z\right)}^{-4}=\dfrac{1}{{\left(-7z\right)}^{4}}[/latex]
• [latex]{t}^{-11}\cdot {t}^{6}[/latex]
• [latex]\dfrac{{25}^{12}}{{25}^{13}}[/latex]
• [latex]{t}^{-5}=\dfrac{1}{{t}^{5}}[/latex]
• [latex]\dfrac{1}{25}[/latex]

Finding the Power of a Product

A general note: the power of a product rule of exponents, example: using the power of a product rule.

• [latex]{\left(a{b}^{2}\right)}^{3}[/latex]
• [latex]{\left(2t\right)}^{15}[/latex]
• [latex]{\left(-2{w}^{3}\right)}^{3}[/latex]
• [latex]\dfrac{1}{{\left(-7z\right)}^{4}}[/latex]
• [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex]
• [latex]{\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}[/latex]
• [latex]2{t}^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[/latex]
• [latex]{\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}[/latex]
• [latex]\dfrac{1}{{\left(-7z\right)}^{4}}=\dfrac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\dfrac{1}{2,401{z}^{4}}[/latex]
• [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\dfrac{{f}^{14}}{{e}^{14}}[/latex]
• [latex]{\left({g}^{2}{h}^{3}\right)}^{5}[/latex]
• [latex]{\left(5t\right)}^{3}[/latex]
• [latex]{\left(-3{y}^{5}\right)}^{3}[/latex]
• [latex]\dfrac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}[/latex]
• [latex]{\left({r}^{3}{s}^{-2}\right)}^{4}[/latex]
• [latex]{g}^{10}{h}^{15}[/latex]
• [latex]125{t}^{3}[/latex]
• [latex]-27{y}^{15}[/latex]
• [latex]\dfrac{1}{{a}^{18}{b}^{21}}[/latex]
• [latex]\dfrac{{r}^{12}}{{s}^{8}}[/latex]

Finding the Power of a Quotient

A general note: the power of a quotient rule of exponents, example: using the power of a quotient rule.

• [latex]{\left(\dfrac{4}{{z}^{11}}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{p}{{q}^{3}}\right)}^{6}[/latex]
• [latex]{\left(\dfrac{-1}{{t}^{2}}\right)}^{27}[/latex]
• [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex]
• [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{4}{{z}^{11}}\right)}^{3}=\dfrac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\dfrac{64}{{z}^{11\cdot 3}}=\dfrac{64}{{z}^{33}}[/latex]
• [latex]{\left(\dfrac{p}{{q}^{3}}\right)}^{6}=\dfrac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\dfrac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\dfrac{{p}^{6}}{{q}^{18}}[/latex]
• [latex]{\left(\dfrac{-1}{{t}^{2}}\right)}^{27}=\dfrac{{\left(-1\right)}^{27}}{{\left({t}^{2}\right)}^{27}}=\dfrac{-1}{{t}^{2\cdot 27}}=\dfrac{-1}{{t}^{54}}=-\dfrac{1}{{t}^{54}}[/latex]
• [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\dfrac{{j}^{3}}{{k}^{2}}\right)}^{4}=\dfrac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\dfrac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\dfrac{{j}^{12}}{{k}^{8}}[/latex]
• [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\dfrac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\dfrac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\dfrac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\dfrac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\dfrac{1}{{m}^{6}{n}^{6}}[/latex]
• [latex]{\left(\dfrac{{b}^{5}}{c}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{5}{{u}^{8}}\right)}^{4}[/latex]
• [latex]{\left(\dfrac{-1}{{w}^{3}}\right)}^{35}[/latex]
• [latex]{\left({p}^{-4}{q}^{3}\right)}^{8}[/latex]
• [latex]{\left({c}^{-5}{d}^{-3}\right)}^{4}[/latex]
• [latex]\dfrac{{b}^{15}}{{c}^{3}}[/latex]
• [latex]\dfrac{625}{{u}^{32}}[/latex]
• [latex]\dfrac{-1}{{w}^{105}}[/latex]
• [latex]\dfrac{{q}^{24}}{{p}^{32}}[/latex]
• [latex]\dfrac{1}{{c}^{20}{d}^{12}}[/latex]

Simplifying Exponential Expressions

Example: simplifying exponential expressions.

• [latex]{\left(6{m}^{2}{n}^{-1}\right)}^{3}[/latex]
• [latex]{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}[/latex]
• [latex]{\left(\dfrac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}[/latex]
• [latex]\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)[/latex]
• [latex]{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}[/latex]
• [latex]\dfrac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}[/latex]
• [latex]\begin{align} {\left(6{m}^{2}{n}^{-1}\right)}^{3}& = {\left(6\right)}^{3}{\left({m}^{2}\right)}^{3}{\left({n}^{-1}\right)}^{3}&& \text{The power of a product rule} \\ & = {6}^{3}{m}^{2\cdot 3}{n}^{-1\cdot 3}&& \text{The power rule} \\ & = 216{m}^{6}{n}^{-3}&& \text{Simplify}. \\ & = \frac{216{m}^{6}}{{n}^{3}}&& \text{The negative exponent rule} \end{align}[/latex]
• [latex]\begin{align} {17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}& =& {17}^{5 - 4-3}&& \text{The product rule} \\ & = {17}^{-2}&& \text{Simplify}. \\ & = \frac{1}{{17}^{2}}\text{ or }\frac{1}{289}&& \text{The negative exponent rule} \end{align}[/latex]
• [latex]\begin{align} {\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}& = \frac{{\left({u}^{-1}v\right)}^{2}}{{\left({v}^{-1}\right)}^{2}}&& \text{The power of a quotient rule} \\ & = \frac{{u}^{-2}{v}^{2}}{{v}^{-2}}&& \text{The power of a product rule} \\ & = {u}^{-2}{v}^{2-\left(-2\right)}&& \text{The quotient rule} \\ & = {u}^{-2}{v}^{4}&& \text{Simplify}. \\ & = \frac{{v}^{4}}{{u}^{2}}&& \text{The negative exponent rule} \end{align}[/latex]
• [latex]\begin{align} \left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)& =& -2\cdot 5\cdot {a}^{3}\cdot {a}^{-2}\cdot {b}^{-1}\cdot {b}^{2}&& \text{Commutative and associative laws of multiplication} \\ & = -10\cdot {a}^{3 - 2}\cdot {b}^{-1+2}&& \text{The product rule} \\ & = -10ab&& \text{Simplify}. \end{align}[/latex]
• [latex]\begin{align} {\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}& = {\left({x}^{2}\sqrt{2}\right)}^{4 - 4} && \text{The product rule} \\ & = {\left({x}^{2}\sqrt{2}\right)}^{0}&& \text{Simplify}. \\ & = 1&& \text{The zero exponent rule} \end{align}[/latex]
• [latex]\begin{align} \frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}& = \frac{{\left(3\right)}^{5}\cdot {\left({w}^{2}\right)}^{5}}{{\left(6\right)}^{2}\cdot {\left({w}^{-2}\right)}^{2}}&& \text{The power of a product rule} \\ & = \frac{{3}^{5}{w}^{2\cdot 5}}{{6}^{2}{w}^{-2\cdot 2}}&& \text{The power rule} \\ & = \frac{243{w}^{10}}{36{w}^{-4}} && \text{Simplify}. \\ & = \frac{27{w}^{10-\left(-4\right)}}{4}&& \text{The quotient rule and reduce fraction} \\ & = \frac{27{w}^{14}}{4}&& \text{Simplify}. \end{align}[/latex]
• [latex]{\left(2u{v}^{-2}\right)}^{-3}[/latex]
• [latex]{x}^{8}\cdot {x}^{-12}\cdot x[/latex]
• [latex]{\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}[/latex]
• [latex]\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)[/latex]
• [latex]{\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3}[/latex]
• [latex]\dfrac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}[/latex]
• [latex]\dfrac{{v}^{6}}{8{u}^{3}}[/latex]
• [latex]\dfrac{1}{{x}^{3}}[/latex]
• [latex]\dfrac{{e}^{4}}{{f}^{4}}[/latex]
• [latex]\dfrac{27r}{s}[/latex]
• [latex]\dfrac{16{h}^{10}}{49}[/latex]

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Cc licensed content, original.

• Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution .

CC licensed content, Shared previously

• College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution . License terms: Download for free at http://cnx.org/contents/ [email protected] .
• Simplify Expressions With Zero Exponents. Authored by: James Sousa (Mathispower4u.com). License: Public Domain: No Known Copyright .
• Simplify Expressions With Negative Exponents. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution .
• Power of a Product. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution .
• Power of a Quotient. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution .
• Question ID 44120, 43231. Authored by: Brenda Gardner. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.
• Question ID 7833, 14060. Authored by: Tyler Wallace. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.
• Question ID 109762, 109765. Authored by: Lumen Learning. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.
• Question ID 51959. Authored by: Roy Shahbazian. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.
• Question ID 93393. Authored by: Michael Jenck. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.
• Question ID 14047, 14058, 14059, 14046, 14051, 14056, 14057. Authored by: James Souza. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.
• Question ID 43896. Authored by: Carla Kulinsky. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.

CC licensed content, Specific attribution

• College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/ [email protected] :1/Preface. License: CC BY: Attribution .

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