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$Select the answer which is equivalent to the given expression using your calculator. {:[(-4a^(2))/((3125 b)^((2)/(5)))],[a=4" and "b=243]:} -(64)/(225) (64)/(225) (8)/(225) -(8)/(225)$

Select the answer which is equivalent to the given expression using your calculator. $\newline$ $\begin{array}{c} \frac{-4 a^{2}}{(3125 b)^{\frac{2}{5}}} \\ a=4 \text { and } b=243 \end{array}$ $\newline$ $-\frac{64}{225}$ $\newline$ $\frac{64}{225}$ $\newline$ $\frac{8}{225}$ $\newline$ $-\frac{8}{225}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

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

\newline

\begin{array}{c} \frac{-4 a^{2}}{(3125 b)^{\frac{2}{5}}} \\ a=4 \text { and } b=243 \end{array}

\newline

-\frac{64}{225}

\newline

\frac{64}{225}

\newline

\frac{8}{225}

\newline

-\frac{8}{225}

Substitute values: Substitute the given values of $a$ and $b$ into the expression. $\newline$ We have the expression $(-4a^2) / ((3125b)^{(2/5)})$ and the values $a=4$ and $b=243$ . $\newline$ Substitute $a=4$ and $b=243$ into the expression.
Calculate numerator: Calculate the numerator. $\newline$ The numerator is $-4a^2$ . Substitute $a=4$ into this to get $-4(4)^2$ . $\newline$ $-4(4)^2 = -4(16) = -64$ .
Calculate denominator: Calculate the denominator. $\newline$ The denominator is $(3125b)^{\frac{2}{5}}$ . Substitute $b=243$ into this to get $(3125\times243)^{\frac{2}{5}}$ . $\newline$ First, calculate $3125\times243$ . $\newline$ $3125\times243 = 759375$ . $\newline$ Now, take the fifth root of $759375$ and then square it, as indicated by the exponent $(\frac{2}{5})$ . $\newline$ (759375)^{\frac{2}{5}} = (759375)^{\frac{1}{5}}^2. $\newline$ Using a calculator, find the fifth root of $759375$ . $\newline$ The fifth root of $759375$ is $b=243$ $0$ . $\newline$ Now square $b=243$ $0$ to get $b=243$ $2$ . $\newline$ $b=243$ $3$ .
Divide numerator by denominator: Divide the numerator by the denominator. $\newline$ Now we have $-64$ for the numerator and $2025$ for the denominator. $\newline$ Divide $-64$ by $2025$ . $\newline$ $-64 / 2025 = -(64/2025)$ .
Simplify fraction: Simplify the fraction if possible. $\newline$ The fraction $-\frac{64}{2025}$ cannot be simplified further as $64$ and $2025$ do not have common factors other than $1$ .