Which expressions are equivalent to $\newline$ $\frac{5^{2}}{5^{8}}$ ? $\newline$ Choose $2$ answers: $\newline$ A $\frac{1}{5^{6}}$ $\newline$ B $1^{-6}$ $\newline$ C $(5^{2})^{-3}$ $\newline$ D $(5^{2})^{-8}$

Full solution

Q. Which expressions are equivalent to

\newline

\frac{5^{2}}{5^{8}}

\newline

Choose

2

answers:

\newline

\frac{1}{5^{6}}

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1^{-6}

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(5^{2})^{-3}

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(5^{2})^{-8}

Simplify expression using laws of exponents: Simplify the expression $(5^{2})/(5^{8})$ by using the laws of exponents. $\newline$ When dividing powers with the same base, subtract the exponents: $5^{2-8} = 5^{-6}$ .
Compare simplified expression with choices: Compare the simplified expression $5^{-6}$ with the given choices. $\newline$ A. $(1)/(5^{6})$ is equivalent to $5^{-6}$ because $1$ divided by any number is the reciprocal of that number, which is expressed as a negative exponent.
Check choice B: Check choice B, which is $1^{(-6)}$ . $\newline$ $1$ raised to any power is $1$ , so $1^{(-6)} = 1$ , which is not equivalent to $5^{(-6)}$ .
Check choice C: Check choice C, which is $(5^{2})^{(-3)}$ . $\newline$ Using the laws of exponents, $(a^{b})^{c} = a^{(b*c)}$ , so $(5^{2})^{(-3)} = 5^{(2*(-3))} = 5^{-6}$ , which is equivalent to the original expression.
Check choice D: Check choice D, which is $(5^{2})^{(-8)}$ . $\newline$ Using the laws of exponents, $(a^{b})^{c} = a^{(b*c)}$ , so $(5^{2})^{(-8)} = 5^{(2*(-8))} = 5^{-16}$ , which is not equivalent to the original expression.