Simplify. ((3z^(2))/(z^(-3)))^(-2) Write your answer using only positive exponents.

Apply Negative Exponent Rule: Apply the negative exponent rule to the entire expression. The negative exponent rule states that a^(-n) = 1/(a^n). We will apply this rule to the entire expression. ((3z^(2))/(z^(-3)))^(-2) = 1/((3z^(2))/(z^(-3)))^(2)Simplify Fraction Inside Parentheses: Simplify the expression inside the parentheses before raising it to the power of 2. We have a fraction raised to a negative exponent. First, we simplify the fraction by applying the rule that z^(a)/z^(b) = z^(a-b). (3z^(2))/(z^(-3)) = 3z^(2 - (-3)) = 3z^(2 + 3) = 3z^(5)Raise Simplified Expression to Power of 2: Now raise the simplified expression to the power of 2. We have (3z^(5))^2. When raising a product to an exponent, we raise each factor to the exponent separately. (3z^(5))^2 = 3^2 * (z^(5))^2 = 9z^(5*2) = 9z^(10)Apply Negative Exponent Rule to Original Expression: Apply the negative exponent rule to the original expression with the simplified base. Now we take the reciprocal of the simplified expression from Step 3, as indicated by the original negative exponent. 1/((3z^(2))/(z^(-3)))^(2) = 1/(9z^(10))

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Simplify.

((3z^(2))/(z^(-3)))^(-2)
Write your answer using only positive exponents.

Simplify. $\newline$ $\left(\frac{3 z^{2}}{z^{-3}}\right)^{-2}$ $\newline$ Write your answer using only positive exponents.

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Algebra 1

Multiplication with rational exponents

Full solution

Q. Simplify.

\newline

\left(\frac{3 z^{2}}{z^{-3}}\right)^{-2}

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Write your answer using only positive exponents.

Apply Negative Exponent Rule: Apply the negative exponent rule to the entire expression. $\newline$ The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$ . We will apply this rule to the entire expression. $\newline$ $\left(\frac{3z^{2}}{z^{-3}}\right)^{-2} = \frac{1}{\left(\frac{3z^{2}}{z^{-3}}\right)^{2}}$
Simplify Fraction Inside Parentheses: Simplify the expression inside the parentheses before raising it to the power of $2$ . We have a fraction raised to a negative exponent. First, we simplify the fraction by applying the rule that $z^{a}/z^{b} = z^{a-b}$ . $(3z^{2})/(z^{-3}) = 3z^{2 - (-3)} = 3z^{2 + 3} = 3z^{5}$
Raise Simplified Expression to Power of $2$ : Now raise the simplified expression to the power of $2$ . $\newline$ We have $(3z^{5})^2$ . When raising a product to an exponent, we raise each factor to the exponent separately. $\newline$ $(3z^{5})^2 = 3^2 \times (z^{5})^2 = 9z^{5\times2} = 9z^{10}$
Apply Negative Exponent Rule to Original Expression: Apply the negative exponent rule to the original expression with the simplified base. $\newline$ Now we take the reciprocal of the simplified expression from Step $3$ , as indicated by the original negative exponent. $\newline$ $\frac{1}{\left(\frac{3z^{2}}{z^{-3}}\right)^{2}} = \frac{1}{9z^{10}}$