Select all the expressions that are equivalent to $5^{-4} \times 6^{-4}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{30^4}$ $\newline$ (B) $\frac{1}{30^{-4}}$ $\newline$ (C) $30^{-4}$ $\newline$ (D) $30^{-8}$

Full solution

Q. Select all the expressions that are equivalent to

5^{-4} \times 6^{-4}

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{30^4}

\newline

(B)

\frac{1}{30^{-4}}

\newline

(C)

30^{-4}

\newline

(D)

30^{-8}

Understand Expression: Understand the given expression $5^{-4} \times 6^{-4}$ . We need to find which of the given choices are equivalent to this expression. To do this, we will simplify the expression using the properties of exponents.
Simplify Expression: Simplify the expression $5^{-4} \times 6^{-4}$ . $\newline$ Since both $5$ and $6$ are raised to the same negative exponent, we can combine them under a single exponent. $\newline$ $5^{-4} \times 6^{-4} = (5\times6)^{-4} = 30^{-4}$
Compare to Choices: Compare the simplified expression to the choices. $\newline$ We have simplified the expression to $30^{-4}$ . Now we need to check which of the given choices match this expression. $\newline$ (A) $\frac{1}{30^4}$ is the same as $30^{-4}$ because a negative exponent indicates the reciprocal of the base raised to the positive exponent. $\newline$ (B) $\frac{1}{30^{-4}}$ is not equivalent because it suggests the reciprocal of $30$ raised to the negative fourth power, which would be $30^4$ . $\newline$ (C) $30^{-4}$ is exactly what we have found in our simplification. $\newline$ (D) $30^{-8}$ is not equivalent because the exponent is $-8$ , not $-4$ . $\newline$ So, the expressions equivalent to $\frac{1}{30^4}$ $0$ are (A) and (C).