Evaluate. ((2)/(3))^(-4)((2)/(3))^(3)

Apply Product Rule: Understand the problem and apply the power of a product rule. We have the expression ((2)/(3))^(-4)((2)/(3))^(3). To simplify this expression, we can use the power of a product rule, which states that when multiplying two expressions with the same base, we can add the exponents.Add Exponents: Add the exponents of the terms with the same base. We have ((2)/(3))^(-4) and ((2)/(3))^(3). Since the base (2/3) is the same, we can add the exponents: -4 + 3. The new exponent is -4 + 3 = -1. So, ((2)/(3))^(-4)((2)/(3))^(3) = ((2)/(3))^(-1).Simplify Negative Exponent: Simplify the expression with the negative exponent. The expression ((2)/(3))^(-1) means the reciprocal of (2/3). So, ((2)/(3))^(-1) = (3/2).

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Evaluate. $\newline$ $\left(\frac{2}{3}\right)^{-4}\left(\frac{2}{3}\right)^{3}$

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Algebra 1

Multiplication with rational exponents

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Q. Evaluate.

\newline

\left(\frac{2}{3}\right)^{-4}\left(\frac{2}{3}\right)^{3}

Apply Product Rule: Understand the problem and apply the power of a product rule. $\newline$ We have the expression $(\frac{2}{3})^{-4}(\frac{2}{3})^{3}$ . To simplify this expression, we can use the power of a product rule, which states that when multiplying two expressions with the same base, we can add the exponents.
Add Exponents: Add the exponents of the terms with the same base. $\newline$ We have $\left(\frac{2}{3}\right)^{-4}$ and $\left(\frac{2}{3}\right)^{3}$ . Since the base $\left(\frac{2}{3}\right)$ is the same, we can add the exponents: $-4 + 3$ . $\newline$ The new exponent is $-4 + 3 = -1$ . $\newline$ So, $\left(\frac{2}{3}\right)^{-4}\left(\frac{2}{3}\right)^{3} = \left(\frac{2}{3}\right)^{-1}$ .
Simplify Negative Exponent: Simplify the expression with the negative exponent. $\newline$ The expression $((2)/(3))^(-1)$ means the reciprocal of $(2/3)$ . $\newline$ So, $((2)/(3))^(-1) = (3/2)$ .