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Select all the expressions that are equivalent to $8^{-2} \times 7^{-2}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{56^2}$ $\newline$ (B) $56^4$ $\newline$ (C) $56^{-2}$ $\newline$ (D) $56^{-4}$

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Simplify rational expressions

Full solution

Q. Select all the expressions that are equivalent to

8^{-2} \times 7^{-2}

.

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{56^2}

\newline

(B)

56^4

\newline

(C)

56^{-2}

\newline

(D)

56^{-4}

Simplify Exponents: First, simplify $8^{-2} \times 7^{-2}$ by using the property of exponents that states $a^{-n} = \frac{1}{a^n}$ . $\newline$ So, $8^{-2} = \frac{1}{8^2} = \frac{1}{64}$ and $7^{-2} = \frac{1}{7^2} = \frac{1}{49}$ . $\newline$ Now, multiply these fractions: $\frac{1}{64} \times \frac{1}{49} = \frac{1}{64 \times 49} = \frac{1}{3136}$ .
Multiply Fractions: Next, compare $\frac{1}{3136}$ with the given choices. $\newline$ (A) $1/56^2 = \frac{1}{56^2} = \frac{1}{3136}$ - This matches our calculation. $\newline$ (B) $56^4$ - This is a positive power, not a negative power or a fraction. $\newline$ (C) $56^{-2} = \frac{1}{56^2} = \frac{1}{3136}$ - This also matches our calculation. $\newline$ (D) $56^{-4} = \frac{1}{56^4}$ - This does not match our calculation.

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