Which expression is equivalent to ((4)/(4^(-2)))^(-6)? 4^(-12) 4^(-18) 4^(-3) 4^(12)

Simplify Exponent Property: Simplify the expression inside the parentheses. We have the expression ((4)/(4^(-2))). To simplify this, we can use the property of exponents that states a^(-n) = 1/(a^n). Therefore, 4^(-2) is equivalent to 1/(4^2). So, ((4)/(4^(-2))) simplifies to (4)/(1/(4^2)) which is the same as 4 * (4^2).Multiply Same Base: Continue simplifying the expression inside the parentheses. Now we multiply 4 by 4^2. When multiplying with the same base, we add the exponents. The base is 4, and the exponents are 1 (for 4, which is 4^1) and 2 (for 4^2). So, 4 * 4^2 becomes 4^(1+2) which is 4^3.Apply Exponent Rule: Apply the exponent outside the parentheses. Now we have (4^3)^(-6). When raising a power to a power, we multiply the exponents. The base is 4, the first exponent is 3, and the second exponent is -6. So, (4^3)^(-6) becomes 4^(3*(-6)) which is 4^(-18).Check Answer Choices: Check the answer choices to see which one matches our result. We have simplified the expression to 4^(-18). Looking at the answer choices, we see that 4^(-18) is one of the options.

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Which expression is equivalent to
((4)/(4^(-2)))^(-6)?

4^(-12)

4^(-18)

4^(-3)

4^(12)

Which expression is equivalent to $\left(\frac{4}{4^{-2}}\right)^{-6} ?$ $\newline$ $4^{-12}$ $\newline$ $4^{-18}$ $\newline$ $4^{-3}$ $\newline$ $4^{12}$

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Algebra 1

Multiplication with rational exponents

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Q. Which expression is equivalent to

\left(\frac{4}{4^{-2}}\right)^{-6} ?

\newline

4^{-12}

\newline

4^{-18}

\newline

4^{-3}

\newline

4^{12}

Simplify Exponent Property: Simplify the expression inside the parentheses. $\newline$ We have the expression $\left(\frac{4}{4^{-2}}\right)$ . To simplify this, we can use the property of exponents that states $a^{-n} = \frac{1}{a^n}$ . Therefore, $4^{-2}$ is equivalent to $\frac{1}{4^2}$ . $\newline$ So, $\left(\frac{4}{4^{-2}}\right)$ simplifies to $\frac{4}{\frac{1}{4^2}}$ which is the same as $4 \times (4^2)$ .
Multiply Same Base: Continue simplifying the expression inside the parentheses. $\newline$ Now we multiply $4$ by $4^2$ . When multiplying with the same base, we add the exponents. The base is $4$ , and the exponents are $1$ (for $4$ , which is $4^1$ ) and $2$ (for $4^2$ ). $\newline$ So, $4 \times 4^2$ becomes $4^{1+2}$ which is $4^2$ $0$ .
Apply Exponent Rule: Apply the exponent outside the parentheses. $\newline$ Now we have $(4^3)^{-6}$ . When raising a power to a power, we multiply the exponents. The base is $4$ , the first exponent is $3$ , and the second exponent is $-6$ . $\newline$ So, $(4^3)^{-6}$ becomes $4^{3*(-6)}$ which is $4^{-18}$ .
Check Answer Choices: Check the answer choices to see which one matches our result. $\newline$ We have simplified the expression to $4^{(-18)}$ . Looking at the answer choices, we see that $4^{(-18)}$ is one of the options.