Algebra II : Simplifying Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #451 : Exponents

\displaystyle 9x\cdot (2x)^3

Possible Answers:

\displaystyle 72x^4

\displaystyle 54x^4

\displaystyle 68x^2

\displaystyle 72x^3

\displaystyle -72x^4

Correct answer:

\displaystyle 72x^4

Explanation:

First, simplify the second expression. When there is an exponent outside the parentheses, it must be distributed to every term in the parentheses:

\displaystyle (2x)^3=8x^3.

Then multiply the two terms:

\displaystyle 9x\cdot 8x^3=72x^4.

Example Question #122 : Multiplying And Dividing Exponents

Divide the exponents:  \displaystyle \frac{2^7}{4^{10}}

Possible Answers:

\displaystyle (\frac{1}{2})^{13}

\displaystyle 4^4

\displaystyle 8

\displaystyle \frac{1}{8}

\displaystyle \frac{1}{2^{13}}

Correct answer:

\displaystyle \frac{1}{2^{13}}

Explanation:

In order to divide the exponents, we must have the same bases. 

Convert the base 4 to 2 by rewriting the four as two squared.

\displaystyle \frac{2^7}{4^{10}}=\frac{2^7}{2^{2(10)}}= \frac{2^7}{2^{20}}

Now that the bases are the same, subtract the power on the numerator with the denominator.

\displaystyle 2^{7-20}= 2^{-13} = \frac{1}{2^{13}}

The answer is:  \displaystyle \frac{1}{2^{13}}

Example Question #452 : Exponents

Simplify: \displaystyle 9^9\div9^{-9}

Possible Answers:

\displaystyle 9

\displaystyle 1

\displaystyle 9^{-18}

\displaystyle 9^{-81}

\displaystyle 9^{18}

Correct answer:

\displaystyle 9^{18}

Explanation:

When dividing exponents with the same base, we subtract the exponents while keeping the base the same.

\displaystyle 9^9\div9^{-9}=9^{9-(-9)}=9^{18}

Example Question #124 : Multiplying And Dividing Exponents

Simplify:

\displaystyle \frac{64x^{-3}y^5z^9}{80x^2y^{-2}z^3}

Possible Answers:

\displaystyle \frac{y^7z^6}{5x^5}

\displaystyle \frac{2y^7z^6}{5x^5}

\displaystyle \frac{4y^7z^6}{5x^5}

\displaystyle \frac{4y^7z^6}{x^5}

\displaystyle \frac{4y^7z^3}{5x^5}

Correct answer:

\displaystyle \frac{4y^7z^6}{5x^5}

Explanation:

To simplify a fraction like this, I look at the like terms separately and simplify those, putting everything together at the end.

First,

\displaystyle \frac{64}{80}=\frac{4}{5}.

Then,

\displaystyle \frac{x^{-3}}{x^2}=\frac{1}{x^5}.

\displaystyle \frac{y^{5}}{y^{-2}}=y^7.

\displaystyle \frac{z^9}{z^3}=z^6.

Put those all together to get your answer:

\displaystyle \frac{4y^7z^6}{5x^5}.

Example Question #125 : Multiplying And Dividing Exponents

\displaystyle 4x\cdot (-2x^3)\cdot (x^{-2})

Possible Answers:

\displaystyle -8x^2

\displaystyle -2x^2

\displaystyle -8x

\displaystyle 8x^2

\displaystyle -8x^3

Correct answer:

\displaystyle -8x^2

Explanation:

Multiply the like terms. Remember that when multiplying with exponents and bases are the same, add exponents.

Therefore,

\displaystyle 4(-2)=-8

and 

\displaystyle x(x^3)(x^{-2})=x^2.

Thus, your answer is:

\displaystyle -8x^2.

Example Question #453 : Exponents

Simplify:

\displaystyle \frac{x^7y^4+x^8y^3+x^3y^7+x^3}{x^3y^3}

Possible Answers:

\displaystyle x^1y^4+y^5+x^4+y^{-3}

\displaystyle x^4y^1+x^5+y^4+y^{-3}

\displaystyle x^4y^1+x^5+y^4+y^{3}

None of these

Correct answer:

\displaystyle x^4y^1+x^5+y^4+y^{-3}

Explanation:

When dividing variables with exponents, the exponents are subtracted

Thus

\displaystyle \frac{x^7y^4+x^8y^3+x^3y^7+x^3}{x^3y^3}

becomes

\displaystyle x^4y^1+x^5+y^4+y^{-3}

 

 

Example Question #124 : Multiplying And Dividing Exponents

Simplify: \displaystyle 49^8\div7^{-8}

Possible Answers:

\displaystyle 7^{-64}

\displaystyle 2

\displaystyle \frac{1}{7}

\displaystyle \frac{1}{2}

\displaystyle 7^{24}

Correct answer:

\displaystyle 7^{24}

Explanation:

Although the bases are not the same, we know that \displaystyle 49=7^2. Therefore, \displaystyle 49^8=(7^2)^8=7^{16}. With the same bases, we now can subtract the exponents while keeping the base the same.

\displaystyle 7^{16}\div7^{-8}=7^{16-(-8)}=7^{24}

Example Question #454 : Exponents

Multiply the following exponents:  \displaystyle 4^{30}\cdot 16^{12}

Possible Answers:

\displaystyle 4^{132}

\displaystyle 4^{54}

\displaystyle 20^{42}

\displaystyle 64^{18}

Correct answer:

\displaystyle 4^{54}

Explanation:

Notice that we can rewrite the second term in terms of the base of four.  Sixteen is equal to four squared.

\displaystyle 16= 4^2

We can replace this term with 16 in order to multiply, and then add the exponents.

\displaystyle 4^{30}\cdot 16^{12} =4^{30}\cdot 4^{2(12)} =4^{30}\cdot 4^{24}

According to the rule of exponents, whenever powers of a similar base are multiplied, the exponents can be added.

\displaystyle x^A \cdot x^B = x^{( A + B )}

The answer is:  \displaystyle 4^{54}

Example Question #129 : Multiplying And Dividing Exponents

Divide:   \displaystyle \frac{4^{34}}{4^{36}}

Possible Answers:

\displaystyle \frac{1}{16}

\displaystyle 1

\displaystyle 4^{70}

\displaystyle 16

\displaystyle 8

Correct answer:

\displaystyle \frac{1}{16}

Explanation:

When dividing powers of a similar base, the exponents can be subtracted.

\displaystyle \frac{4^{34}}{4^{36}} = 4^{34-36}=4^{-2}

Simplify the negative exponent.

\displaystyle x^{-n} = \frac{1}{x^n}

\displaystyle 4^{-2} = \frac{1}{4^2} = \frac{1}{16}

The answer is:  \displaystyle \frac{1}{16}

Example Question #130 : Multiplying And Dividing Exponents

Evaluate:  \displaystyle \frac{3^{15}}{27^{18}}

Possible Answers:

\displaystyle \frac{1}{3^9}

\displaystyle \frac{1}{9^{117}}

\displaystyle \frac{1}{9^3}

\displaystyle \frac{1}{9^{39}}

\displaystyle \frac{1}{3^{39}}

Correct answer:

\displaystyle \frac{1}{3^{39}}

Explanation:

In order to simplify this, we will need to rewrite the base 27 as a common base of three in order to be able to subtract the exponents.

\displaystyle 27 = 3^3

Rewrite the fraction.

\displaystyle \frac{3^{15}}{27^{18}} = \frac{3^{15}}{3^{3(18)}} = \frac{3^{15}}{3^{54}}

Subtract the exponents.

\displaystyle 3^{15-54} = 3^{-39}=\frac{1}{3^{39}}

The answer is:  \displaystyle \frac{1}{3^{39}}

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