Select the equivalent expression. ((1)/(k^(-8)*k^(5)))^((5)/(3)) (1)/(k^(5)) k^(5) (1)/(root(5)(k)) root(5)(k)

Simplify Exponents: Simplify the expression inside the parentheses by adding the exponents of k. When multiplying powers with the same base, we add the exponents. So, k^(-8) * k^(5) = k^(-8 + 5) = k^(-3). The expression becomes ((1)/(k^(-3)))^((5)/(3)).Apply Quotient Rule: Apply the power of a quotient rule. The power of a quotient rule states that (a/b)^n = a^n / b^n. So, ((1)/(k^(-3)))^((5)/(3)) = 1^((5)/(3)) / (k^(-3))^((5)/(3)).Simplify Numerator: Simplify the numerator. Any number raised to any power is itself if the number is 1. So, 1^((5)/(3)) = 1. The expression now is 1 / (k^(-3))^((5)/(3)).Apply Power Rule: Apply the power to a power rule. The power to a power rule states that (a^n)^m = a^(n*m). So, (k^(-3))^((5)/(3)) = k^(-3 * (5)/(3)) = k^(-5). The expression now is 1 / k^(-5).Move Base to Numerator: Simplify the expression by moving the k term to the numerator. When we have a negative exponent, we can move the base to the opposite side of the fraction to make the exponent positive. So, 1 / k^(-5) = k^(5).

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Select the equivalent expression.

((1)/(k^(-8)*k^(5)))^((5)/(3))

(1)/(k^(5))

k^(5)

(1)/(root(5)(k))

root(5)(k)

Select the equivalent expression. $\newline$ $\left(\frac{1}{k^{-8} \cdot k^{5}}\right)^{\frac{5}{3}}$ $\newline$ $\frac{1}{k^{5}}$ $\newline$ $k^{5}$ $\newline$ $\frac{1}{\sqrt[5]{k}}$ $\newline$ $\sqrt[5]{k}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Select the equivalent expression.

\newline

\left(\frac{1}{k^{-8} \cdot k^{5}}\right)^{\frac{5}{3}}

\newline

\frac{1}{k^{5}}

\newline

k^{5}

\newline

\frac{1}{\sqrt[5]{k}}

\newline

\sqrt[5]{k}

Simplify Exponents: Simplify the expression inside the parentheses by adding the exponents of $k$ . When multiplying powers with the same base, we add the exponents. So, $k^{-8} \times k^{5} = k^{-8 + 5} = k^{-3}$ . The expression becomes $\left(\frac{1}{k^{-3}}\right)^{\frac{5}{3}}$ .
Apply Quotient Rule: Apply the power of a quotient rule. $\newline$ The power of a quotient rule states that $(a/b)^n = a^n / b^n$ . $\newline$ So, $((1)/(k^{-3}))^{(5)/(3)} = 1^{(5)/(3)} / (k^{-3})^{(5)/(3)}$ .
Simplify Numerator: Simplify the numerator. $\newline$ Any number raised to any power is itself if the number is $1$ . $\newline$ So, $1^{(5/3)} = 1$ . $\newline$ The expression now is $1 / (k^{-3})^{(5/3)}$ .
Apply Power Rule: Apply the power to a power rule. $\newline$ The power to a power rule states that $(a^n)^m = a^{n*m}$ . $\newline$ So, $(k^{-3})^{(5)/(3)} = k^{-3 * (5)/(3)} = k^{-5}$ . $\newline$ The expression now is $1 / k^{-5}$ .
Move Base to Numerator: Simplify the expression by moving the $k$ term to the numerator. $\newline$ When we have a negative exponent, we can move the base to the opposite side of the fraction to make the exponent positive. $\newline$ So, $1 / k^{-5} = k^{5}$ .