Select the equivalent expression. (2^(-4)*z^(-3))^(5)=? Choose 1 answer: (A) 2z^(2) (B) 2^(20)*z^(15) (c) (1)/(2^(20)*z^(15))

Apply power rule: Apply the power of a power rule. The power of a power rule states that (a^m)^n = a^(m*n). We will apply this rule to both 2^(-4) and z^(-3). (2^(-4)*z^(-3))^5 = 2^(-4*5) * z^(-3*5)Perform exponent multiplication: Perform the multiplication for the exponents. Now we multiply the exponents by 5. 2^(-4*5) * z^(-3*5) = 2^(-20) * z^(-15)Write with positive exponents: Write the expression using positive exponents. Since we have negative exponents, we can write them as fractions with positive exponents in the denominator. 2^(-20) * z^(-15) = (1/2^20) * (1/z^15)Combine fractions: Combine the fractions. Since both terms are being multiplied, we can combine them into a single fraction. (1/2^20) * (1/z^15) = 1/(2^20 * z^15)Match with options: Match the result with the given options. The expression we have found is 1/(2^20 * z^15), which matches option (C).

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Math Problems

Grade 8

Evaluate expressions using properties of exponents

Full solution

Q. Select the equivalent expression.

\newline

(2^{-4} \cdot z^{-3})^{5}=

\newline

Choose

1

answer:

\newline

(A)

2z^{2}

\newline

(B)

2^{20} \cdot z^{15}

\newline

(C)

\frac{1}{2^{20} \cdot z^{15}}

Apply power rule: Apply the power of a power rule. $\newline$ The power of a power rule states that $(a^m)^n = a^{(m*n)}$ . We will apply this rule to both $2^{-4}$ and $z^{-3}$ . $\newline$ $(2^{-4}*z^{-3})^5 = 2^{(-4*5)} * z^{(-3*5)}$
Perform exponent multiplication: Perform the multiplication for the exponents. $\newline$ Now we multiply the exponents by $5$ . $\newline$ $2^{(-4*5)} * z^{(-3*5)} = 2^{-20} * z^{-15}$
Write with positive exponents: Write the expression using positive exponents. $\newline$ Since we have negative exponents, we can write them as fractions with positive exponents in the denominator. $\newline$ $2^{-20} \times z^{-15} = \frac{1}{2^{20}} \times \frac{1}{z^{15}}$
Combine fractions: Combine the fractions. $\newline$ Since both terms are being multiplied, we can combine them into a single fraction. $\newline$ $(\frac{1}{2^{20}}) \times (\frac{1}{z^{15}}) = \frac{1}{2^{20} \times z^{15}}$
Match with options: Match the result with the given options. $\newline$ The expression we have found is $\frac{1}{2^{20} \cdot z^{15}}$ , which matches option (C).