Distributing Exponents (Power Rule) - Algebra 2

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Question

Simplify:

$\left(\frac{1}{3}^7\right)^{-7}$

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Answer

When an exponent is raised by another exponent, we will multiply the exponents and keep the base the same.

Simplify:

$\left(\frac{1}{3}^7\right)^{-7}=\frac{1}{3}^{7*-7}=\frac{1}{3}^{-49}$

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Question

Simplify:

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Answer

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

$(8^8)^8=8^{8*8}=8^{64}$

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Question

Simplify: $(9^9)^{-9}$

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Answer

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

$(9^9)^{-9}=9^{-9*9}=9^{-81}$

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Question

Simplify: $(11^{-11})^{-11}$

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Answer

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

$(11^{-11})^{-11}=11^{-11*-11}=11^{121}$

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Question

Simplify:

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Answer

Although we don't have the same base, we know that . Therefore

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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Question

Simplify: $144^6\div12^7$

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Answer

Although we don't have the same base, we know that . 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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Question

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

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Answer

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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Question

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

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Answer

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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Question

Simplify:

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

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Answer

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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Question

Simplify:

$(5^{-8})^{-12}$

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Answer

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

$(5^{-8})^{-12}=5^{-8*-12}=5^{96}$

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Question

Simplify: $(3a^2b^{-3}c^4)^2$

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Answer

To simplify this expression, square every term in the parentheses:

$3^2a^4b^{-6}c^8$ .

Then simplify and get rid of the negative exponent by putting the b term on the denominator:

$\frac{9a^4c^8}{b^6}$ .

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Question

Simplify:

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Answer

When an exponent is raised by another exponent, we will multiply the exponents and keep the base the same.

Simplify:

$(4^7)^7=4^{7*7}=4^{49}$

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Question

Simplify:

$(8^7)^{-2}$

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Answer

When an exponent is raised by another exponent, we will multiply the exponents and keep the base the same.

Simplify:

$(8^7)^{-2}=8^{7*-2}=8^{-14}$

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Question

Simplify:

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

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Answer

When an exponent is raised by another exponent, we will multiply the exponents and keep the base the same.

Simplify:

$(5^{-5})^{-5}=5^{-5*-5}=5^{25}$

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Question

What is the largest positive integer, , such that is a factor of ?

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Answer

. Thus, is equal to 16.

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Question

Solve:

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Answer

Solve each term separately. A number to the zeroth power is equal to 1, but be careful to apply the signs after the terms have been simplified.

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Question

Evaluate:

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Answer

When dealing exponents being raised by a power, we multiply the exponents and keep the base.

$(2^9)^4=2^{9*4}=2^{36}$

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Question

Evaluate:

$(5^{6})^7$

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Answer

When dealing exponents being raised by a power, we multiply the exponents and keep the base.

$(5^{6})^7=5^{6*7}=5^{42}$

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Question

Evaluate:

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Answer

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

$(9^9)^9=9^{9*9}=9^{81}$

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Question

Simplify the expression:

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

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Answer

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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