Select all the expressions that are equivalent to $2^{-1} \times 2^{-4}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{2^{-5}}$ $\newline$ (B) $\frac{1}{2^{5}}$ $\newline$ (C) $2^{-5}$ $\newline$ (D) $2^{4}$

Full solution

Q. Select all the expressions that are equivalent to

2^{-1} \times 2^{-4}

\newline

Multi-select Choices:

\newline

(A)

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

\newline

(B)

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

\newline

(C)

2^{-5}

\newline

(D)

2^{4}

Simplify Exponents: Simplify the expression $2^{-1} \times 2^{-4}$ by adding the exponents. $\newline$ When multiplying powers with the same base, we add the exponents. $\newline$ $2^{-1} \times 2^{-4} = 2^{(-1 + -4)} = 2^{-5}$
Compare Simplified Expression: Compare the simplified expression to the choices given. $\newline$ We have simplified the expression to $2^{-5}$ . Now we need to determine which of the given choices are equivalent to this expression.
Evaluate Choice (A): Evaluate choice (A) $\frac{1}{2^{-5}}$ . The expression $\frac{1}{2^{-5}}$ is equivalent to $2^5$ because a negative exponent indicates that the base should be on the other side of the fraction line. $\frac{1}{2^{-5}} = 2^5$ This is not equivalent to $2^{-5}.$
Evaluate Choice (B): Evaluate choice (B) $\frac{1}{2^5}$ . The expression $\frac{1}{2^5}$ is the reciprocal of $2^5$ , which is equivalent to $2^{-5}$ . $\frac{1}{2^5} = 2^{-5}$ This is equivalent to $2^{-5}$ .
Evaluate Choice (C): Evaluate choice (C) $2^{-5}$ . The expression $2^{-5}$ is exactly the same as our simplified expression. $2^{-5} = 2^{-5}$ This is equivalent to $2^{-5}.$
Evaluate Choice (D): Evaluate choice (D) $2^4$ . The expression $2^4$ is the reciprocal of $2^{-4}$ , which is not equivalent to $2^{-5}$ . $2^4 \neq 2^{-5}$ This is not equivalent to $2^{-5}$ .