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High School Math : Simplifying Exponents

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Example Questions

Example Question #1 : Simplifying Exponents

Simplify the following expression.

$\frac{4x^3y^8z^2}{8x^6y^2z^4}$

Possible Answers:

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

$\frac{y^4}{2x^2z^2}$

$\frac{4y^4}{8x^2z^2}$

$\frac{4y^6}{8x^3z^2}$

Correct answer:

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

Explanation:

When dividing with exponents, the exponent in the denominator is subtracted from the exponent in the numerator. For example: $\frac{x^3}{x^6}= x^{3-6}=x^{-3}=\frac{1}{x^3}$ .

In our problem, each term can be treated in this manner. Remember that a negative exponent can be moved to the denominator.

$\frac{4x^3y^8z^2}{8x^6y^2z^4}=\frac{4}{8}x^{-3}y^6z^{-2}=\frac{4y^6}{8x^3z^2}$

Now, simplifly the numerals.

$\frac{4y^6}{8x^3z^2}=\frac{y^6}{2x^3z^2}$

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Example Question #1 : Multiplying And Dividing Exponents

Simplify the following expression.

$2^{3} \cdot 2^{6}$

Possible Answers:

$2^{9}$

$4^{18}$

$2^{18}$

$6^{6}$

$4^{9}$

Correct answer:

$2^{9}$

Explanation:

We are given: $2^{3} \cdot 2^{6}$ .

Recall that when we are multiplying exponents with the same base, we keep the base the same and add the exponents.

Thus, we have $2^{3 + 6} = 2^{9}$ .

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Example Question #1 : Multiplying And Dividing Exponents

Simplify the following expression.

$\frac{2^{9}}{2^{11}}$

Possible Answers:

$2^{\frac{9}{11}}$

$\frac{18}{22}$

$2^{2}$

$\frac{9}{11}$

$\frac{1}{2^{2}}$

Correct answer:

$\frac{1}{2^{2}}$

Explanation:

Recall that when we are dividing exponents with the same base, we keep the base the same and subtract the exponents.

Thus, we have $\frac{2^{9}}{2^{11}} = 2^{9 - 11} = 2^{-2}$ .

We also recall that for negative exponents,

$b^{-x} = \frac{1}{b^{x}}$ .

Thus, $2^{-2} = \frac{1}{2^{2}}$ .

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Example Question #1 : Multiplying And Dividing Exponents

Simplify the following exponent expression:

$(\frac{a^{\frac{-3}{2}}}{3^6b^{\frac{-2}{3}}})^{\frac{-1}{2}}$

Possible Answers:

$\frac{a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{27a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{9a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{3a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{18a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

Correct answer:

$\frac{27a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

Explanation:

Begin by rearranging the terms in the numerator and denominator so that the exponents are positive:

$(\frac{a^{\frac{-3}{2}}}{3^6b^{\frac{-2}{3}}})^{\frac{-1}{2}}$

$(\frac{3^6b^{\frac{-2}{3}}}{a^{\frac{-3}{2}}})^{\frac{1}{2}}$

$(\frac{3^6a^{\frac{3}{2}}}{b^{\frac{2}{3}}})^{\frac{1}{2}}$

Multiply the exponents:

$(\frac{3^3a^{\frac{3}{4}}}{b^{\frac{1}{3}}})$

Simplify:

$\frac{27a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

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Example Question #1 : Simplifying Exponents

Simplify the expression:

$x^3 + \frac{2x^2}{x^{-1}}$

Possible Answers:

3x^3

x^3+2x

x^3+2x^2

$x^3+\frac{2}{x^3}$

x^3+2

Correct answer:

3x^3

Explanation:

First simplify the second term, and then combine the two:

$x^3 + \frac{2x^2}{x^{-1}}$

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

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Example Question #1 : Multiplying And Dividing Exponents

Solve for : 4^y = 32*2^2^y^-^x

Possible Answers:

x=5

x=3

x=0

Cannot be determined from the given information.

x=-3

Correct answer:

x=5

Explanation:

Rewrite each side of the equation to only use a base 2:

$2^{2y} = 2^{5}*2^{2y-x}$

$2^{2y} = 2^{5+2y-x}$

The only way this equation can be true is if the exponents are equal.

So:

2y = 5+2y-x

The on each side cancel, and moving the to the left side, we get:

x=5

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Example Question #4 : Simplifying Exponents

$\log_{x}27e^{3}=3$

Solve for .

Possible Answers:

1/2e

Correct answer:

Explanation:

First, set up the equation: x^3=27 e^3 . Simplifying this result gives x = 3 .

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Example Question #2 : Simplifying Exponents

What is the largest positive integer, , such that 2^z is a factor of 16^4 ?

Possible Answers:

Correct answer:

Explanation:

16^4 = (2^4)^4 = 2^1^6 . Thus, is equal to 16.

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Example Question #1 : Simplifying Exponents

Order the following from least to greatest:

$25^{100}$

$2^{300}$

$3^{500}$

$4^{400}$

$2^{600}$

Possible Answers:

$2^{300}, 2^{600}, 3^{500}, 25^{100}, 4^{400}$

$2^{300}, 25^{100}, 2^{600}, 3^{500}, 4^{400}$

$25^{100}, 2^{300}, 2^{600}, 3^{500}, 4^{400}$

$3^{500}, 4^{400}, 2^{300}, 25^{100}, 2^{600}$

$2^{600}, 2^{300}, 25^{100}, 3^{500}, 4^{400}$

Correct answer:

$2^{300}, 25^{100}, 2^{600}, 3^{500}, 4^{400}$

Explanation:

In order to solve this problem, each of the answer choices needs to be simplified.

Instead of simplifying completely, make all terms into a form such that they have 100 as the exponent. Then they can be easily compared.

2^3^0^0 = (2^3)^1^0^0 = 8^1^0^0 , 3^5^0^0 = (3^5)^1^0^0 = 243^1^0^0 , 4^4^0^0 = (4^4)^1^0^0 = 256^1^0^0 , and 2^6^0^0 = (2^6)^1^0^0 = 64^1^0^0 .

Thus, ordering from least to greatest: 8^1^0^0, 25^1^0^0, 64^1^0^0, 243^1^0^0, 256^1^0^0 .

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Example Question #1 : Review And Other Topics

Simplify the expression:

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

Possible Answers:

$\frac{3x}{16y^3}$

$\frac{3x}{64y^4}$

$\frac{3x}{64y^3}$

Cannot be simplified

Correct answer:

$\frac{3x}{64y^4}$

Explanation:

Begin by distributing the exponent through the parentheses. The power rule dictates that an exponent raised to another exponent means that the two exponents are multiplied:

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

Any negative exponents can be converted to positive exponents in the denominator of a fraction:

$\frac{3x^4y^2}{64x^3y^6}$

The like terms can be simplified by subtracting the power of the denominator from the power of the numerator:

$\frac{3x^4y^2}{64x^3y^6}=\frac{3}{64}x^{4-3}y^{2-6}=\frac{3}{64}xy^{-4}$

$\frac{3x}{64y^4}$

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High School Math : Simplifying Exponents

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Simplifying Exponents

Example Question #1 : Multiplying And Dividing Exponents

Example Question #1 : Multiplying And Dividing Exponents

Example Question #1 : Multiplying And Dividing Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Multiplying And Dividing Exponents

Example Question #4 : Simplifying Exponents

Example Question #2 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Review And Other Topics

All High School Math Resources

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