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Algebra II : Distributing Exponents (Power Rule)

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Example Question #231 : Simplifying Exponents

Simplify: $144^6\div12^7$

Possible Answers:

$12^{-3}$

12^6

$12^{-1}$

$12^{-2}$

12^5

Correct answer:

12^5

Explanation:

Although we don't have the same base, we know that 144=12^2 . Therefore we have $144^6\div12^7=(12^2)^6\div12^7=12^{12}\div12^7$

When dividing exponents with the same base, we subtract the exponents and keep the base the same.

$12^{12}\div12^7=12^{12-7}=12^5$

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Example Question #51 : Distributing Exponents (Power Rule)

Simplify: $(19^\frac{2}{3})^9$

Possible Answers:

19^2

$19^{\frac{1}{2}}$

$19^{\frac{4}{7}}$

$19^{18}$

19^6

Correct answer:

19^6

Explanation:

When dealing with exponents raising another exponent, we just multiply the powers and keep the base the same.

$(19^\frac{2}{3})^9=19^{\frac{2}{3}*9}=19^6$

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Example Question #233 : Simplifying Exponents

Simplify $(x^{-6}x^{4})^{3}$ .

Possible Answers:

$\frac{1}{x^{6}}$

$x^{6}$

$\frac{1}{x^{72}}$

$\frac{1}{x^{30}}$

Correct answer:

$\frac{1}{x^{6}}$

Explanation:

The problem uses both the Power of a Product Property and the Power of a Power Property of exponents.

$x^{a}x^{b}=x^{a+b}$

$(x^{a})^{b}=x^{ab}$

First multiply the terms inside the parentheses:

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

Now raise the term to the 3rd power:

$(x^{-2})^{3}=x^{-2\cdot 3}=x^{-6}$

Negative exponents mean that the term belongs in the denominator:

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

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Example Question #51 : Distributing Exponents (Power Rule)

Simplify: 6^7*36^3

Possible Answers:

$6^{42}$

$216^{10}$

$216^{42}$

$6^{13}$

$6^{10}$

Correct answer:

$6^{13}$

Explanation:

Although we don't have the same base, we know that 36=6^2 . Therefore 6^7*36^3=6^7*(6^2)^3=6^7*6^6

When multiplying exponents with the same base, we just add the exponents and keep the base the same.

$6^7*6^6=6^{6+7}=6^{13}$

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Example Question #52 : Distributing Exponents (Power Rule)

Simplify: (3x^2y^5)^4

Possible Answers:

$81x^8y^{20}$

$\textup{The answer is not given.}$

81x^6y^9

$81x^{16}y^{625}$

$12x^8y^{20}$

Correct answer:

$81x^8y^{20}$

Explanation:

When the terms are braced and are raised to a power of four, this is similar to multiplying the quantified terms by itself four times.

(3x^2y^5)^4 = (3x^2y^5)(3x^2y^5)(3x^2y^5)(3x^2y^5)

This also means that the powers inside the brace can be multiplied the power outside of the parentheses.

$3^{1\cdot 4}x^{2\cdot 4}y^{5\cdot 4} = 3^4x^8y^{20} = 81x^8y^{20}$

The answer is: $81x^8y^{20}$

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Example Question #53 : Distributing Exponents (Power Rule)

Simplify:

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

Possible Answers:

$\frac{-8y^5}{x^6}$

$\frac{8y^9}{x^6}$

$\frac{-8y^9}{x^3}$

$\frac{y^9}{x^6}$

$\frac{-8y^9}{x^6}$

Correct answer:

$\frac{-8y^9}{x^6}$

Explanation:

The power that's outside of the parentheses needs to be distributed to every term inside the parentheses:

$-2^3x^{-2\cdot 3}y^{3\cdot 3}$ .

When there's a power outside the parentheses, the exponents are multiplied:

$-8x^{-6}y^9$ .

To get rid of the negative exponent, put it on the denominator:

$\frac{-8y^9}{x^6}$ .

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Example Question #54 : Distributing Exponents (Power Rule)

Simplify: $2^4\cdot 8^{10}$

Possible Answers:

$2^{34}$

$2^{120}$

$4^{13}$

$2^{42}$

$8^{22}$

Correct answer:

$2^{34}$

Explanation:

Because the bases are not common, we cannot apply the power rule as is.

We will need to convert the to a common base in order to rewrite the exponents.

The base eight is the same as two to the power of three.

Rewrite the second term with a base of two.

$2^4\cdot 8^{10}=2^4\cdot 2^{3(10)}$

Simplify the exponents. Now that the bases are common, they can be combined as one exponent.

$2^4\cdot 2^{3(10)} = 2^4\cdot 2^{30} = 2^{34}$

The correct answer is: $2^{34}$

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Example Question #55 : Distributing Exponents (Power Rule)

Evaluate: 2(a^3b^4)^5

Possible Answers:

$2a^{15}b^{20}$

$32a^{15}b^{20}$

$10a^{15}b^{20}$

$2a^{8}b^{9}$

$32a^{8}b^{9}$

Correct answer:

$2a^{15}b^{20}$

Explanation:

The product rule for exponents can be applied when there is an exponent applied to a quantity.

Multiply the powers together for each variable and then distribute the integer. Do not raise the integer to the power of five since it is not inside the parentheses.

$2(a^3b^4)^5 = 2(a^{3\cdot 5}b^{4\cdot 5 }) =2a^{15}b^{20}$

The answer is: $2a^{15}b^{20}$

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Example Question #52 : Distributing Exponents (Power Rule)

Simplify: (7^7)^7

Possible Answers:

49^7

$7^{49}$

$7^{14}$

$14^{49}$

$7^{77}$

Correct answer:

$7^{49}$

Explanation:

When dealing with exponents being raised by another exponent, we just multiply the powers and keep the base the same.

$(7^7)^7=7^{7*7}=7^{49}$

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Example Question #53 : Distributing Exponents (Power Rule)

Simplify: $(2^5)^{12}$

Possible Answers:

$2^{17}$

8^6

$2^{30}$

$4^{15}$

$2^{60}$

Correct answer:

$2^{60}$

Explanation:

When dealing with exponents being raised by another exponent, we just multiply the powers and keep the base the same.

$(2^5)^{12}=2^{5*12}=2^{60}$

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Algebra II : Distributing Exponents (Power Rule)

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #231 : Simplifying Exponents

Example Question #51 : Distributing Exponents (Power Rule)

Example Question #233 : Simplifying Exponents

Example Question #51 : Distributing Exponents (Power Rule)

Example Question #52 : Distributing Exponents (Power Rule)

Example Question #53 : Distributing Exponents (Power Rule)

Example Question #54 : Distributing Exponents (Power Rule)

Example Question #55 : Distributing Exponents (Power Rule)

Example Question #52 : Distributing Exponents (Power Rule)

Example Question #53 : Distributing Exponents (Power Rule)

All Algebra II Resources

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