Select all the expressions that are equivalent to $(7^1)^{-1}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{7}$ $\newline$ (B) $7^0$ $\newline$ (C) $\frac{1}{7^{-1}}$ $\newline$ (D) $7^{-1}$

Full solution

Q. Select all the expressions that are equivalent to

(7^1)^{-1}

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{7}

\newline

(B)

7^0

\newline

(C)

\frac{1}{7^{-1}}

\newline

(D)

7^{-1}

Simplify Exponent: Simplify the expression $(7^1)^{-1}$ . When an exponent is raised to another exponent, we multiply the exponents. Since $7^1$ is simply $7$ , we have: $(7^1)^{-1} = 7^{1 \cdot -1} = 7^{-1}$
Compare to Choices: Compare the simplified expression to the choices. $\newline$ We have determined that $7^1)^{-1}\ is equivalent to \$7^{-1}$ . Now we need to see which of the choices are equivalent to $7^{-1}$ . $\newline$ (A) $\frac{1}{7}$ is the same as $7^{-1}$ because a negative exponent indicates the reciprocal of the base raised to the positive exponent. $\newline$ (B) $7^0$ is not equivalent to $7^{-1}$ because any non-zero number raised to the power of $0$ is $1$ . $\newline$ (C) $\frac{1}{7^{-1}}$ is equivalent to $7$ because the negative exponent in the denominator moves the base to the numerator and makes the exponent positive. $\newline$ (D) $7^{-1}$ is clearly equivalent to $7^{-1}$ by definition.
Select Correct Expressions: Select all the correct expressions. $\newline$ From our comparison in Step $2$ , we can see that the expressions equivalent to $(7^1)^{-1}$ are: $\newline$ (A) $\frac{1}{7}$ $\newline$ (D) $7^{-1}$ $\newline$ Choices (B) and (C) are not equivalent to $(7^1)^{-1}$ .