Select the equivalent expression. root(30)((1)/(k^(-7)*k^(-5))) k^(-(2)/(5)) k^(-(5)/(2)) k^((5)/(2)) k^((2)/(5))

Simplify Inside Radical: We are given the expression: root(30)((1)/(k^(-7)*k^(-5))) First, we need to simplify the expression inside the radical.Combine Exponents: Since the bases are the same (k), we can add the exponents when multiplying: k^(-7) * k^(-5) = k^(-7 + -5) = k^(-12) So, the expression becomes: root(30)((1)/(k^(-12)))Move to Numerator: Now, we can rewrite the expression by moving k to the numerator and changing the sign of the exponent: root(30)(k^12)Rewrite Radical as Exponent: The radical root(30) can be rewritten as an exponent of 1/30: (k^12)^(1/30)Multiply Exponents: When raising a power to a power, we multiply the exponents: (k^12)^(1/30) = k^(12 * 1/30) = k^(12/30)Simplify Fraction: We can simplify the fraction 12/30 by dividing both the numerator and the denominator by their greatest common divisor, which is 6: k^(12/30) = k^(2/5)

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Select the equivalent expression.

root(30)((1)/(k^(-7)*k^(-5)))

k^(-(2)/(5))

k^(-(5)/(2))

k^((5)/(2))

k^((2)/(5))

Select the equivalent expression. $\newline$ $\sqrt[30]{\frac{1}{k^{-7} \cdot k^{-5}}}$ $\newline$ $k^{-\frac{2}{5}}$ $\newline$ $k^{-\frac{5}{2}}$ $\newline$ $k^{\frac{5}{2}}$ $\newline$ $k^{\frac{2}{5}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Select the equivalent expression.

\newline

\sqrt[30]{\frac{1}{k^{-7} \cdot k^{-5}}}

\newline

k^{-\frac{2}{5}}

\newline

k^{-\frac{5}{2}}

\newline

k^{\frac{5}{2}}

\newline

k^{\frac{2}{5}}

Simplify Inside Radical: We are given the expression: $\newline$ $\sqrt[30]{\frac{1}{k^{-7} \cdot k^{-5}}}$ $\newline$ First, we need to simplify the expression inside the radical.
Combine Exponents: Since the bases are the same $k$ , we can add the exponents when multiplying: $k^{-7} \times k^{-5} = k^{-7 + -5} = k^{-12}$ So, the expression becomes: $\sqrt[30]{\frac{1}{k^{-12}}}$
Move to Numerator: Now, we can rewrite the expression by moving $k$ to the numerator and changing the sign of the exponent: $\sqrt[30]{k^{12}}$
Rewrite Radical as Exponent: The radical $\sqrt[30]{k^{12}}$ can be rewritten as an exponent of $\frac{1}{30}$ : $k^{12^{\frac{1}{30}}}$
Multiply Exponents: When raising a power to a power, we multiply the exponents: k^{$12$})^{\frac{$1$}{$30$}} = k^{$12$ \times \frac{$1$}{$30$}} = k^{\frac{$12$}{$30$}}\
Simplify Fraction: We can simplify the fraction \(\frac{12}{30} by dividing both the numerator and the denominator by their greatest common divisor, which is $6$ : $k^{\frac{12}{30}} = k^{\frac{2}{5}}$