Evaluate. 32^((-3)/(5))=2^◻

Evaluate Base: Evaluate the base of the exponent to see if it can be related to the exponent's form. 32 is a power of 2, specifically 2^5, since 2 * 2 * 2 * 2 * 2 = 32.Rewrite Expression: Rewrite the expression using the base of 2. 32^((-3)/(5)) can be written as (2^5)^((-3)/(5)).Apply Power Rule: Apply the power of a power rule, which states that (a^b)^c = a^(b*c). (2^5)^((-3)/(5)) becomes 2^(5 * (-3)/(5)).Simplify Exponent: Simplify the exponent by multiplying 5 by -3/5. 5 * (-3)/(5) = -3.Negative Exponent: Now the expression is 2^(-3). To simplify a negative exponent, recall that a^(-n) = 1/(a^n). 2^(-3) = 1/(2^3).Calculate Value: Calculate 2^3. 2^3 = 2 * 2 * 2 = 8.Substitute Back: Substitute the value back into the expression. 2^(-3) = 1/(2^3) = 1/8.

Home

Math Problems

Grade 7

Powers with negative bases

Full solution

Q. Evaluate.

\newline

32^{\frac{-3}{5}}=2^{\square}

Evaluate Base: Evaluate the base of the exponent to see if it can be related to the exponent's form. $\newline$ $32$ is a power of $2$ , specifically $2^5$ , since $2 \times 2 \times 2 \times 2 \times 2 = 32$ .
Rewrite Expression: Rewrite the expression using the base of $2$ . $32^{(-3)/(5)}$ can be written as $(2^5)^{(-3)/(5)}$ .
Apply Power Rule: Apply the power of a power rule, which states that a^b)^c = a^{(b*c)}\. $\(2^ $5$ )^{( $-3$ )/( $5$ )}\ becomes (2$^{( $5$ * ( $-3$ )/( $5$ ))}\.
Simplify Exponent: Simplify the exponent by multiplying $5$ by $-\frac{3}{5}$ . $5 \times \left(-\frac{3}{5}\right) = -3$ .
Negative Exponent: Now the expression is $2^{-3}$ . $\newline$ To simplify a negative exponent, recall that $a^{-n} = \frac{1}{a^n}$ . $\newline$ $2^{-3} = \frac{1}{2^3}.$
Calculate Value: Calculate $2^3$ . $2^3 = 2 \times 2 \times 2 = 8$ .
Substitute Back: Substitute the value back into the expression. $\newline$ $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ .