Select all the expressions that are equivalent to $(2^{-6})^1$ . $\newline$ Multi-select Choices: $\newline$ (A) $2^{-5}$ $\newline$ (B) $\frac{1}{2^6}$ $\newline$ (C) $\frac{1}{2^{-5}}$ $\newline$ (D) $2^{-6}$

Full solution

Q. Select all the expressions that are equivalent to

(2^{-6})^1

\newline

Multi-select Choices:

\newline

(A)

2^{-5}

\newline

(B)

\frac{1}{2^6}

\newline

(C)

\frac{1}{2^{-5}}

\newline

(D)

2^{-6}

Simplify Exponent: Simplify the expression $(2^{-6})^1$ . When an exponent is raised to another exponent, we multiply the exponents. In this case, the exponent $1$ does not change the value of the base or its exponent, so $(2^{-6})^1$ simplifies to $2^{-6}$ . Calculation: $(2^{-6})^1 = 2^{(-6*1)} = 2^{-6}$
Calculate Simplified Expression: Compare the simplified expression to the choices given. $\newline$ We have determined that $(2^{-6})^1$ simplifies to $2^{-6}$ . Now we need to compare this to the choices given to see which ones are equivalent. $\newline$ (A) $2^{-5}$ is not equivalent because the exponent $-5$ is not the same as $-6$ . $\newline$ (B) $\frac{1}{2^6}$ is equivalent because $2^{-6}$ can be written as $\frac{1}{2^6}$ . $\newline$ (C) $\frac{1}{2^{-5}}$ is not equivalent because the negative exponent in the denominator implies a positive exponent in the numerator, which is not the same as $2^{-6}$ . $\newline$ (D) $2^{-6}$ is equivalent because it is the same expression we started with.