lesson 5 problem solving practice negative exponents answers

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Negative exponents

Here you will learn about negative exponents, including how to simplify and evaluate with negative exponents.

Students will first learn about negative exponents as part of expressions and equations in 8 th grade, and will continue to expand their knowledge through high school.

What are negative exponents?

Negative exponents are powers (also called indices) with a negative sign (minus sign) in front of them.

Examples of negative exponents:

2b^{-\frac{1}{2}}

You get negative exponents by dividing two terms with the same base where the first term is raised to a power that is smaller than the power that the second term is raised to. Similarly to how a positive exponent means repeated multiplication, a negative exponent means repeated division.

For example,

When you cancel the common factors of x,

You are left with,

Using the division law of exponents, you know that,

How to use negative exponents

A negative exponent can be defined as the multiplicative inverse of the base raised to the power, which is of the opposite sign of the given power.

In other words, in order to make the negative exponent positive, put the term over 1 and flip it. It is known as finding the reciprocal of the base (term).

The negative exponent rule states that a number with a negative exponent should be put in the denominator.

is the same as

Negative exponents will often be used in conjunction with other exponent laws, including division, parentheses, and multiplication laws.

You can also figure out the value of negative number expressions by identifying patterns. For example, notice how when the power decreases by 1, the answer is half of the answer of the previous expression.

What do you think 2^{-2} will be equal to?

Common Core State Standards

How does this relate to 8 th grade math?

Grade 8: Expressions and Equations (8.EE.A.1) Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 \times 3-5 = 3-3 = \cfrac{1}{33} = \cfrac{1}{27}.

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How to evaluate negative exponents

In order to evaluate a negative exponent, you need to:

Put the term over \bf{1}.

Flip the fraction to make the exponent positive.

Simplify, if necessary.

Negative exponents examples

Example 1: no coefficient in front of the base.

Simplify and leave your answer in index form.

When the exponent is negative, put it over 1,

2 Flip the fraction to make the exponent positive.

Flip and change the power from -4 to +4.

Example 2: negative exponents

7^{-9}=\cfrac{7^{-9}}{1}

Flip and change the power from -9 to 9.

=\cfrac{1}{7^9}

Example 3: with a coefficient in front of base

Notice how the exponent affects the entire bracket.

\cfrac{(10 a)^{-3}}{1}

Flip and change the power from -3 to +3.

\cfrac{1}{(10 a)^{3}}

Simplify the denominator.

=\cfrac{1}{1000 a^{3}}

Example 4: with a coefficient in front of base

Notice how the exponent only affects the variable b.

\begin{aligned}3 b^{-2} &=3 \times b^{-2} \\\\ &=3 \times \cfrac{b^{-2}}{1}\end{aligned}

Flip and change the power -2 to +2.

\begin{aligned}3 \times \cfrac{b^{-2}}{1} &=3 \times \cfrac{1}{b^{2}} \\\\ &=\cfrac{3}{b^{2}}\end{aligned}

The exponent only applies to the variable b and not the coefficient 3.

Example 5: with fractional exponents

When dealing with fractions, skip to step 2.

Flip and change the power from -2 to +2.

\left(\cfrac{4}{3}\right)^{-2}=\left(\cfrac{3}{4}\right)^{2}

Simplify the numerator and denominator.

\begin{aligned}&=\cfrac{3^{2}}{4^{2}} \\\\ &=\cfrac{9}{16}\end{aligned}

Example 6: with fractional exponents

\left(\cfrac{14}{4}\right)^{-2}=\left(\cfrac{4}{14}\right)^2

\begin{aligned}& =\cfrac{4^2}{14^2} \\\\ & =\cfrac{16}{196} \\\\ & =\cfrac{4}{49} \end{aligned}

Teaching tips for negative exponents

Introduce the concept using concrete examples that can illustrate it. Using simple numerical examples can make it easy to show how negative exponents relate to taking the reciprocal.
Offer a variety of practice problems to reinforce the concept. This can be done using a worksheet, but find an interactive way to allow students to discuss and problem solve together.
Allow students the opportunity to explore real-life applications of negative exponents in fields such as science, engineering, or finance.

Easy mistakes to make

Confusing integer and fractional powers Raising a term to the power of 2 means you square it. For example, 2^{2}=2 \times 2 Raising a term to the power of \cfrac{1}{2} means we find the square root of it. For example, 2^{\frac{1}{2}}=\pm \sqrt{2}
Thinking indices, powers or exponents are all different Exponents can also be called powers or indices.
Not turning a negative exponent into a positive exponent when flipping the term After putting the term over one, make sure to flip the exponent from a negative to a positive exponent.
Making a mistake when writing one over a fraction Typically, when finding the reciprocal, you write the term over one and flip the term. However, when you are finding the reciprocal of a fraction, the steps are a little different. This is why when dealing with fractions it is easier to skip to changing the powers and just flip the fraction.

Related laws of exponents lessons

Exponential notation
Dividing exponents
Multiplying exponents
Square root
Exponent rules

Practice negative exponents questions

1. Simplify. Express your answer in index form.

The negative exponent means finding the reciprocal, so

2. Simplify. Express your answer in index form.

3. Simplify. Express your answer in index form.

The negative exponent means finding the reciprocal, but this only applies to the variable, so

4. Evaluate. Express your answer in index form.

The negative exponent means finding the reciprocal, which means inverting the fraction, so

5. Evaluate. Express your answer in index form.

6. Evaluate. Express your answer in index form.

Negative exponents FAQs

Yes, the base can be a positive number, negative number, or zero. The negative exponent only affects the power to which the base is raised to.

Scientific notation involves negative exponents often. They can represent numbers in the form of a \times 10^{-n}, where a is a number between 1 and 10.

A positive power results in the multiplication of the base by itself multiple times, while negative powers result in taking the reciprocal of the base raised to the corresponding positive exponent.

Yes, according to the properties of exponents, the power of zero, any nonzero base raised to the zero power equals 1.

Some of the fundamental rules of exponents are: ∘ Product rule: \left(a^m \times a^n\right)=a^{m+n} ∘ Quotient rule: \cfrac{a^m}{a^n}=a^{m-n} ∘ Power rule: \left(a^m\right)^n=a^{m \times n}

The next lessons are

Radical functions
Math formulas
Quadratic graphs

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3.7: Negative Exponents

Last updated
Save as PDF
Page ID 58091

Denny Burzynski & Wade Ellis, Jr.
College of Southern Nevada via OpenStax CNX

Reciprocals

Negative exponents, working with negative exponents.

Two real numbers are said to be reciprocals of each other if their product is 1. Every nonzero real number has exactly one reciprocal, as shown in the examples below. Zero has no reciprocal.

Example $\PageIndex{1}$

$4 \cdot \dfrac{1}{4} = 1$. This means that $4$ and $\dfrac{1}{4}$ are reciprocals.

Example $\PageIndex{2}$

$6 \cdot \dfrac{1}{6} = 1$. This means that $6$ and $\dfrac{1}{6}$ are reciprocals.

Example $\PageIndex{3}$

$-2 \cdot \dfrac{-1}{2} = 1$. This means that $-2$ and $-\dfrac{1}{2}$ are reciprocals.

Example $\PageIndex{4}$

$a \cdot \dfrac{1}{a} = 1$. This means that $a$ and $\dfrac{1}{a}$ are reciprocals.

Example $\PageIndex{5}$

$x \cdot \dfrac{1}{x} = 1$. This means that $x$ and $\dfrac{1}{x}$ are reciprocals.

Example $\PageIndex{6}$

$x^3 \cdot \dfrac{1}{x^3} = 1$. This means that $x^3$ and $\dfrac{1}{x^3}$ are reciprocals.

We can use the idea of reciprocals to find a meaning for negative exponents.

Consider the product of $x^3$ and $x^{-3}$. Assume $x \not = 0$.

\[x^3 \cdot x^{-3} = x^{3 + (-3)} = x^0 = 1\]

Thus, since the product of $x^3$ and $x^{-3}$ is $1$, $x^3$ and $x^{-3}$ must be reciprocals.

We also know that $x^3 \cdot \dfrac{1}{x^3} = 1$. (See problem 6 above.) Thus, $x^3$ and $\dfrac{1}{x^3}$ are also reciprocals.

Then, since $x^{-3}$ and $\dfrac{1}{x^3}$ are both reciprocals of $x^3$ and a real number can have only one reciprocal, it must be that $x^{-3} = \dfrac{1}{x^3}$.

We have used $-3$ as the exponent, but the process works as well for all other negative integers. We make the following definition.:

Negative Exponent Definition

If $n$ is any natural number and $x$ is any nonzero real number, then:

$x^{-n} = \dfrac{1}{x^n}$

Sample Set A

Write each of the following so that only positive exponents appear.

Example $\PageIndex{7}$

$x^{-6} = \dfrac{1}{x^6}$

Example $\PageIndex{8}$

$a^{-1} = \dfrac{1}{a^a} = \dfrac{1}{a}$

Example $\PageIndex{9}$

$7^{-2} = \dfrac{1}{7^2} = \dfrac{1}{49}$

Example $\PageIndex{10}$

$(3a)^{-6} = \dfrac{1}{(3a)^6}$

Example $\PageIndex{11}$

$(5x-1)^{-24} = \dfrac{1}{(5x-1)^{-24}}$

Example $\PageIndex{12}$

$(k+2x)^{-(-8)} = (k+2z)^8$

Practice Set A

Write each of the following using only positive exponents.

Practice Problem $\PageIndex{1}$

$\dfrac{1}{y^5}$

Practice Problem $\PageIndex{2}$

$\dfrac{1}{m^2}$

Practice Problem $\PageIndex{3}$

$\dfrac{1}{9}$

Practice Problem $\PageIndex{4}$

$\dfrac{1}{5}$

Practice Problem $\PageIndex{5}$

$\dfrac{1}{16}$

Practice Problem $\PageIndex{6}$

$(xy)^{-4}$

$\dfrac{1}{(xy)^4}$

Practice Problem $\PageIndex{7}$

$(a+2b)^{-12}$

$\dfrac{1}{(a+2b)^{12}}$

Practice Problem $\PageIndex{8}$

$(m-n)^{-(-4)}$

$(m-n)^4$

It is important to note that $a^{-n}$ is not necessarily a negative number. For example,

$3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ $3^{-2} \not = \ -9$

The problems of Sample Set A suggest the following rule for working with exponents:

Moving Factors Up and Down

In a fraction, a factor can be moved from the numerator to the denominator or from the denominator to the numerator by changing the sign of the exponent.

Sample Set B

Example $\pageindex{13}$.

$x^{-2}y^5$.

The factor $x^{-2}$ can be moved from the numerator to the denominator by changing the exponent $-2$ to $+2$

$x^{-2}y^5 = \dfrac{y^5}{x^2}$

Example $\PageIndex{14}$

$a^9b^{-3}$.

The factor $b^{-3}$ can be moved from the numerator to the denominator by changing the exponent $-3$ to $+3$.

$a^9b^{-3} = \dfrac{a^9}{b^3}$

Example $\PageIndex{15}$

$\dfrac{a^4b^2}{c^{-6}}$.

This fraction can be written without any negative exponents by moving the factor $c^{-6}$ into the numerator. We must change the $-6$ to $+6$ to make the move legitimate.

$\dfrac{a^4b^2}{c^{-6}} = a^4b^2c^6$

Example $\PageIndex{16}$

$\dfrac{1}{x^{-3}y^{-2}z^{-1}}$.

This fraction can be written without negative exponents by moving all the factors from the denominator to the numerator. Change the sign of each exponent: $-3$ to $+3$, $-2$ to $+2$, $-1$ to $+1$.

$\dfrac{1}{x^{-3}y^{-2}z^{-1}} = x^3y^2z^1 = x^3y^2z$

Practice Set B

Practice problem $\pageindex{9}$.

$x^{-4}y^7$

$\dfrac{y^7}{x^4}$

Practice Problem $\PageIndex{10}$

$\dfrac{a^2}{b^{-4}}$

Practice Problem $\PageIndex{11}$

$\dfrac{x^3y^4}{z^{-8}}$

$x^3y^4z^8$

Practice Problem $\PageIndex{12}$

$\dfrac{6m^{-3}n^{-2}}{7k^{-1}}$

$\dfrac{6k}{7m^3n^2}$

Practice Problem $\PageIndex{13}$

$\dfrac{1}{a^{-2}b^{-6}c^{-8}}$

$a^2b^6c^8$

Practice Problem $\PageIndex{14}$

$\dfrac{3a(a-5b)^{-2}}{5b(a-4b)^5}$

$\dfrac{3a}{5b(a-5b)^2(a-4b)^5}$

Sample Set C

Example $\pageindex{17}$.

Rewrite $\dfrac{24 a^{7} b^{9}}{2^{3} a^{4} b^{-6}}$ in a simpler form. Notice that we are dividing powers with the same base. We'll proceed by using the rules of exponents. $ \begin{aligned} \dfrac{24 a^{7} b^{9}}{2^{3} a^{4} b^{-6}}=\dfrac{24 a^{7} b^{9}}{8 a^{4} b^{-6}} &=3 a^{7-4} b^{9-(-6)} \\ &=3 a^{3} b^{9+6} \\ &=3 a^{3} b^{15} \end{aligned} $

Write $\dfrac{9 a^{5} b^{3}}{5 x^{3} y^{2}}$ so that no denominator appears. We can eliminate the denominator by moving all factors that make up the denominator to the numerator. $9 a^{5} b^{3} 5^{-1} x^{-3} y^{-2}$

Find the value of $\dfrac{1}{10^{-2}}+\dfrac{3}{4^{-3}}$ We can evaluate this expression by eliminating the negative exponents. $ \begin{aligned} \dfrac{1}{10^{-2}}+\dfrac{3}{4^{-3}} &=1 \cdot 10^{2}+3 \cdot 4^{3} \\ &=1 \cdot 100+3 \cdot 64 \\ &=100+192 \\ &=292 \end{aligned} $

Practice Set C

Practice problem $\pageindex{15}$.

Rewrite $\dfrac{36x^8b^3}{3^2x^{-2}b^{-5}}$ in a simpler form.

$4x^{10}b^8$

Practice Problem $\PageIndex{16}$

Write $\dfrac{2^4m^{-3}n^7}{4^{-1}x^5}$ in a simpler form and one in which no denominator appears.

$64m^{-3}n^7x^{-5}$

Practice Problem $\PageIndex{17}$

Find the value of $\dfrac{2}{5^{-2}} + 6^{-2} \cdot 2^3 \cdot 3^2$

Write the following expressions using only positive exponents. Assume all variables are nonzero.

Exercise $\PageIndex{1}$

$\dfrac{1}{x^{-2}}$

Exercise $\PageIndex{2}$

Exercise $\pageindex{3}$.

$\dfrac{1}{x^7}$

Exercise $\PageIndex{4}$

Exercise $\pageindex{5}$.

$a^{-10}$

$\dfrac{1}{a^{-10}}$

Exercise $\PageIndex{6}$

$b^{-12}$

Exercise $\PageIndex{7}$

$b^{-14}$

$\dfrac{1}{b^{14}}$

Exercise $\PageIndex{8}$

Exercise $\pageindex{9}$, exercise $\pageindex{10}$.

$(x+1)^{-2}$

Exercise $\PageIndex{11}$

$(x-5)^{-3}$

$\dfrac{1}{(x-5)^3}$

Exercise $\PageIndex{12}$

$(y-4)^{-6}$

Exercise $\PageIndex{13}$

$(a+9)^{-10}$

$\dfrac{1}{(a+9)^{10}}$

Exercise $\PageIndex{14}$

$(r+3)^{-8}$

Exercise $\PageIndex{15}$

$(a-1)^{-12}$

$\dfrac{1}{(a-1)^{12}}$

Exercise $\PageIndex{16}$

$x^3y^{-2}$

Exercise $\PageIndex{17}$

$x^7y^{-5}$

$\dfrac{x^7}{y^5}$

Exercise $\PageIndex{18}$

$a^4b^{-1}$

Exercise $\PageIndex{19}$

$a^7b^{-8}$

$\dfrac{a^7}{b^8}$

Exercise $\PageIndex{20}$

$a^2b^3c^{-2}$

Exercise $\PageIndex{21}$

$x^3y^2z^{-6}$

$\dfrac{x^3y^2}{z^6}$

Exercise $\PageIndex{22}$

$x^3y^{-4}z^2w$

Exercise $\PageIndex{23}$

$a^7b^{-9}zw^3$

$\dfrac{a^7zw^3}{b^9}$

Exercise $\PageIndex{24}$

$a^3b^{-1}zw^2$

Exercise $\PageIndex{25}$

$x^5y^{-5}z^{-2}$

$\dfrac{x^5}{y^5z^2}$

Exercise $\PageIndex{26}$

$x^4y^{-8}z^{-3}w^{-4}$

Exercise $\PageIndex{27}$

$a^{-4}b^{-6}c^{-1}d^4$

$\dfrac{d^4}{a^4b^6c}$

Exercise $\PageIndex{28}$

$x^9y^{-6}z^{-1}w^{-5}r^{-2}$

Exercise $\PageIndex{29}$

$4x^{-6}y^2$

$\dfrac{4y^2}{x^6}$

Exercise $\PageIndex{30}$

$5x^2y^2z^{-5}$

Exercise $\PageIndex{31}$

$7a^{-2}b^2c^2$

$\dfrac{7b^2c^2}{a^2}$

Exercise $\PageIndex{32}$

$4x^3(x+1)^2y^{-4}z^{-1}$

Exercise $\PageIndex{33}$

$7a^2(a-4)^3b^{-6}c^{-7}$

$\dfrac{7a^2(a-4)^3}{b^6c^7}$

Exercise $\PageIndex{34}$

$18b^{-6}(b^2-3)^{-5}c^{-4}d^5e^{-1}$

Exercise $\PageIndex{35}$

$7(w+2)^{-2}(w+1)^3$

$\dfrac{7(w+1)^3}{(w+2)^2}$

Exercise $\PageIndex{36}$

$2(a-8)^{-3}(a-2)^5$

Exercise $\PageIndex{37}$

$(x^2+3)^3(x^2-1)^{-4}$

$\dfrac{(x^2+3)^3}{(x^2-1)^4}$

Exercise $\PageIndex{38}$

$(x^4+2x-1)^{-6}(x+5)^4$

Exercise $\PageIndex{39}$

$(3x^2-4x-8)^{-9}(2x+11)^{-3}$

$\dfrac{1}{(3x^2-4x-8)^{9}(2x+11)^{3}}$

Exercise $\PageIndex{40}$

$(5y^2+8y-6)^{-2}(6y-1)^{-7}$

Exercise $\PageIndex{41}$

$7a(a^2-4)^{-2}(b^2-1)^{-2}$

$\dfrac{7a}{(a^2-4)^2(b^2-1)^2}$

Exercise $\PageIndex{42}$

$(x-5)^{-4}3b^2c^4(x+6)^8$

Exercise $\PageIndex{43}$

$(y^3+1)^{-1}5y^3z^{-4}w^{-2}(y^3-1)^{-2}$

$\dfrac{5y^3}{(y^3+1)z^4w^2(y^3-1)^2}$

Exercise $\PageIndex{44}$

$5x^3(2x^{-7})$

Exercise $\PageIndex{45}$

$3y^{-3}(9x)$

$\dfrac{27x}{y^3}$

Exercise $\PageIndex{46}$

$6a^{-4}(2a^{-6})$

Exercise $\PageIndex{47}$

$4a^2b^2a^{-5}b^{-2}$

$\dfrac{4}{a^3}$

Exercise $\PageIndex{48}$

$5^{-1}a^{-2}b^{-6}b^{-11}c^{-3}c^9$

Exercise $\PageIndex{49}$

$2^3x^22^{-3}x^{-2}$

Exercise $\PageIndex{50}$

$7a^{-3}b^{-9} \cdot 5a^6bc^{-2}c^4$

Exercise $\PageIndex{51}$

$(x+5)^2(x+5)^{-6}$

$\dfrac{1}{(x+5)^4}$

Exercise $\PageIndex{52}$

$(a-4)^3(a-4)^{-10}$

Exercise $\PageIndex{53}$

$8(b+2)^{-8}(b+2)^{-4}(b+2)^3$

$\dfrac{8}{(b+2)^9}$

Exercise $\PageIndex{54}$

$3a^5b^{-7}(a^2+4)^{-3}6a^{-4}b(a^2+4)^{-1}(a^2+4)$

Exercise $\PageIndex{55}$

$-4a^3b^{-5}(2a^2b^7c^{-2})$

$\dfrac{-8a^5b^2}{c^2}$

Exercise $\PageIndex{56}$

$-2x^{-2}y^{-4}z^4(-6x^3y^{-3}z)$

Exercise $\PageIndex{57}$

$(-5)^2(-5)^{-1}$

Exercise $\PageIndex{58}$

$(-9)^{-3}(9)^3$

Exercise $\PageIndex{59}$

$(-1)^{-1}(-1)^{-1}$

Exercise $\PageIndex{60}$

$(4)^2(2)^{-4}$

Exercise $\PageIndex{61}$

$\dfrac{1}{a^{-4}}$

Exercise $\PageIndex{62}$

$\dfrac{1}{a^{-1}}$

Exercise $\PageIndex{63}$

$\dfrac{4}{x^{-6}}$

Exercise $\PageIndex{64}$

$\dfrac{7}{x^{-8}}$

Exercise $\PageIndex{65}$

$\dfrac{23}{y^{-1}}$

Exercise $\PageIndex{66}$

$\dfrac{6}{a^2b^{-4}}$

Exercise $\PageIndex{67}$

$\dfrac{3c^5}{a^3b^{-3}}$

$\dfrac{3b^3c^5}{a^3}$

Exercise $\PageIndex{68}$

$\dfrac{16a^{-2}b^{-6}c}{2yz^{-5}w^{-4}}$

Exercise $\PageIndex{69}$

$\dfrac{24y^2z^{-8}}{6a^2b^{-1}c^{-9}d^3}$

$\dfrac{4bc^9y^2}{a^2d^3z^8}$

Exercise $\PageIndex{70}$

$\dfrac{3^{-1}b^5(b+7)^{-4}}{9^{-1}a^{-4}(a+7)^2}$

Exercise $\PageIndex{71}$

$\dfrac{36a^6b^5c^8}{3^2a^3b^7c^9}$

$\dfrac{4a^3}{b^2c}$

Exercise $\PageIndex{72}$

$\dfrac{45a^4b^2c^6}{15a^2b^7c^8}$

Exercise $\PageIndex{73}$

$\dfrac{3^3x^4y^3z}{3^2xy^5z^5}$

$\dfrac{3x^3}{y^2z^4}$

Exercise $\PageIndex{74}$

$\dfrac{21x^2y^2z^5w^4}{7xyz^{12}w^{14}}$

Exercise $\PageIndex{75}$

$\dfrac{33a^{-4}b^{-7}}{11a^3b^{-2}}$

$\dfrac{3}{a^7b^5}$

Exercise $\PageIndex{76}$

$\dfrac{51x^{-5}y^{-3}}{3xy}$

Exercise $\PageIndex{77}$

$\dfrac{2^6x^{-5}y^{-2}a^{-7}b^5}{2^{-1}x^{-4}y^{-2}b^6}$

$\dfrac{128}{a^7bx}$

Exercise $\PageIndex{78}$

$\dfrac{(x+3)^3(y-6)^4}{(x+3)^5(y-6)^{-8}}$

Exercise $\PageIndex{79}$

$\dfrac{4x^3}{y^7}$

Exercise $\PageIndex{80}$

$\dfrac{5x^4y^3}{a^3}$

Exercise $\PageIndex{81}$

$\dfrac{23a^4b^5c^{-2}}{x^{-6}y^5}$

$\dfrac{23a^4b^5x^6}{c^2y^5}$

Exercise $\PageIndex{82}$

$\dfrac{2^3b^5c^2d^{-9}}{4b^4cx}$

Exercise $\PageIndex{83}$

$\dfrac{10x^3y^{-7}}{3x^5z^2}$

$\dfrac{10}{3x^2y^7z^2}$

Exercise $\PageIndex{84}$

$\dfrac{3x^2y^{-2}(x-5)}{9^{-1}(x+5)^3}$

Exercise $\PageIndex{85}$

$\dfrac{14a^2b^2c^{-12}(a^2+21)^{-4}}{4^{-2}a^2b^{-1}(a+6)^3}$

$\dfrac{224b^3}{c^{12}(a^2+21)^4(a+6)^3}$

For the following problems, evaluate each numerical expression.

Exercise $\PageIndex{86}$

Exercise $\pageindex{87}$.

$\dfrac{1}{7}$

Exercise $\PageIndex{88}$

Exercise $\pageindex{89}$.

$\dfrac{1}{32}$

Exercise $\PageIndex{90}$

Exercise $\pageindex{91}$.

$6 \cdot 3^{-3}$

$\dfrac{2}{9}$

Exercise $\PageIndex{92}$

$4 \cdot 9^{-2}$

Exercise $\PageIndex{93}$

$28 \cdot 14^{-1}$

Exercise $\PageIndex{94}$

$2^{-3}(3^{-2})$

Exercise $\PageIndex{95}$

$2^{-1} \cdot 3^{-1} \cdot 4^{-1}$

$\dfrac{1}{24}$

Exercise $\PageIndex{96}$

$10^{-2} + 3(10^{-2})$

Exercise $\PageIndex{97}$

$(-3)^{-2}$

Exercise $\PageIndex{98}$

$(-10)^{-1}$

Exercise $\PageIndex{99}$

$\dfrac{3}{2^{-3}}$

Exercise $\PageIndex{100}$

$\dfrac{4^{-1}}{5^{-2}}$

Exercise $\PageIndex{101}$

$\dfrac{2^4-7}{4^{-1}}$

Exercise $\PageIndex{102}$

$\dfrac{2^{-1}+4^{-1}}{2^{-2} + 4^{-2}}$

Exercise $\PageIndex{103}$

$\dfrac{21^0-2^6}{2 \cdot 6-13}$

For the following problems, write each expression so that only positive exponents appear.

Exercise $\PageIndex{104}$

$(a^6)^{-2}$

Exercise $\PageIndex{105}$

$(a^5)^{-3}$

$\dfrac{1}{a^{15}}$

Exercise $\PageIndex{106}$

$(x^7)^{-4}$

Exercise $\PageIndex{107}$

$(x^4)^{-8}$

$\dfrac{1}{x^{32}}$

Exercise $\PageIndex{108}$

$(b^{-2})^7$

Exercise $\PageIndex{109}$

$(b^{-4})^{-1}$

Exercise $\PageIndex{110}$

$(y^{-3})^{-4}$

Exercise $\PageIndex{111}$

$(y^{-9})^{-3}$

Exercise $\PageIndex{112}$

$(a^{-1})^{-1}$

Exercise $\PageIndex{113}$

$(b^{-1})^{-1}$

Exercise $\PageIndex{114}$

$(a^0)^{-1}$, $a \not = 0$

Exercise $\PageIndex{115}$

$(m^))^{-1}$, $m \not = 0$

Exercise $\PageIndex{116}$

$(x^{-3}y^7)^{-4}$

Exercise $\PageIndex{117}$

$(x^6y^6z^{-1})^2$

$\dfrac{x^{12}y^{12}}{z^2}$

Exercise $\PageIndex{118}$

$(a^{-5}b^{-1}c^0)^6$

Exercise $\PageIndex{119}$

$(\dfrac{y^3}{x^{-4}})^5$

$x^{20}y^{15}$

Exercise $\PageIndex{120}$

$(\dfrac{a^{-8}}{b^{-6}})^3$

Exercise $\PageIndex{121}$

$(\dfrac{2a}{b^3})^4$

$\dfrac{16a^4}{b^{12}}$

Exercise $\PageIndex{122}$

$(\dfrac{3b}{a^2})^{-5}$

Exercise $\PageIndex{123}$

$(\dfrac{5^{-1}a^3b^{-6}}{x^{-2}y^9})^2$

$\dfrac{a^6x^4}{25b^{12}y^{18}}$

Exercise $\PageIndex{124}$

$(\dfrac{4m^{-3}n^6}{2m^{-5}n})^3$

Exercise $\PageIndex{125}$

$(\dfrac{r^5s^{-4}}{m^{-8}n^7})^{-4}$

$\dfrac{n^{28}s^{16}}{m^{32}r^{20}}$

Exercise $\PageIndex{126}$

$(\dfrac{h^{-2}j^{-6}}{k^{-4}p})^{-5}$

Exercises for Review

Exercise $\pageindex{127}$.

Simplify $(4x^5y^3z^0)^3$

$64x^{15}y^9$

Exercise $\PageIndex{128}$

Find the sum. $-15 + 3$

Exercise $\PageIndex{129}$

Find the difference. $8 -(-12)$

Exercise $\PageIndex{130}$

Simplify $(-3)(-8) + 4(-5)$

Exercise $\PageIndex{131}$

Find the value of $m$ if $m = \dfrac{-3k-5t}{kt+6}$ when $k = 4$ and $t = -2$

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Negative Exponents and the Laws of Exponents

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Grade 8 Mathematics Module 1, Topic A, Lesson 5: Student Version
Grade 8 Mathematics Module 1, Topic A, Lesson 5: Teacher Version

Prerequisites

CCSS Standard:

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SAT Mathematics : Solving Problems with Exponents

Study concepts, example questions & explanations for sat mathematics, all sat mathematics resources, example questions, example question #1 : solving problems with exponents.