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$Select the answer which is equivalent to the given expression using your calculator. {:[((-5a)^(2))/((32 b)^((3)/(5)))],[a=-5" and "b=3125]:} (5)/(8) (5)/(16) -(5)/(8) -(5)/(16)$

Select the answer which is equivalent to the given expression using your calculator. $\newline$ $\begin{array}{c} \frac{(-5 a)^{2}}{(32 b)^{\frac{3}{5}}} \\ a=-5 \text { and } b=3125 \end{array}$ $\newline$ $\frac{5}{8}$ $\newline$ $\frac{5}{16}$ $\newline$ $-\frac{5}{8}$ $\newline$ $-\frac{5}{16}$

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Precalculus

Operations with rational exponents

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Q. Select the answer which is equivalent to the given expression using your calculator.

\newline

\begin{array}{c} \frac{(-5 a)^{2}}{(32 b)^{\frac{3}{5}}} \\ a=-5 \text { and } b=3125 \end{array}

\newline

\frac{5}{8}

\newline

\frac{5}{16}

\newline

-\frac{5}{8}

\newline

-\frac{5}{16}

Substitute Values: Substitute the given values of $a$ and $b$ into the expression. $\newline$ Given expression: $\frac{(-5a)^{2}}{(32b)^{\frac{3}{5}}}$ $\newline$ Substitute $a = -5$ and $b = 3125$ . $\newline$ $\frac{(-5\times(-5))^{2}}{(32\times3125)^{\frac{3}{5}}}$
Simplify Numerator: Simplify the numerator. $\newline$ $(-5 \times -5) = 25$ $\newline$ $(25)^{2} = 625$
Simplify Denominator: Simplify the denominator. $\newline$ First, calculate $32 \times 3125$ . $\newline$ $32 \times 3125 = 100000$ $\newline$ Now, take the fifth root of $100000$ and then cube it. $\newline$ $(100000)^{\frac{3}{5}} = (100000)^{\frac{1}{5}})^3$ $\newline$ The fifth root of $100000$ is $10$ because $10^5 = 100000$ . $\newline$ So, $(10)^3 = 1000$
Divide and Convert: Divide the numerator by the denominator. $\newline$ $\frac{625}{1000} = 0.625$ $\newline$ Convert $0.625$ to a fraction. $\newline$ $0.625 = \frac{625}{1000}$ $\newline$ Simplify the fraction by dividing both numerator and denominator by $125$ . $\newline$ $\left(\frac{625}{125}\right) / \left(\frac{1000}{125}\right) = \frac{5}{8}$
Determine Sign: Determine the sign of the result. $\newline$ Since the original expression had an even power on the negative term in the numerator, the result will be positive. $\newline$ So, the final answer is positive $\frac{5}{8}$ .