((1)/(9))^(-(3)/(2))*((1)/(8))^((2)/(3)) What is the value of the given expression?

Understand and Simplify Exponents: Understand the given expression and simplify the exponents. The given expression is ((1)/(9))^(-(3)/(2))*((1)/(8))^((2)/(3)). We need to simplify each part of the expression separately.Simplify ((1)/(9))^(-(3)/(2)): Simplify the first part of the expression ((1)/(9))^(-(3)/(2)). The negative exponent means we take the reciprocal of the base and then apply the positive exponent. So, ((1)/(9))^(-(3)/(2)) becomes (9)^(3/2).Calculate (9)^(3/2): Calculate (9)^(3/2). Since 9 is a perfect square (3^2), we can rewrite (9)^(3/2) as (3^2)^(3/2). When we raise a power to a power, we multiply the exponents, so (3^2)^(3/2) = 3^(2*(3/2)) = 3^3. Now calculate 3^3 = 3 * 3 * 3 = 27.Simplify ((1)/(8))^((2)/(3)): Simplify the second part of the expression ((1)/(8))^((2)/(3)). The exponent (2/3) means we take the cube root of the base and then square it. So, ((1)/(8))^((2)/(3)) becomes (1/2)^2.Calculate (1/2)^2: Calculate (1/2)^2. Square the fraction (1/2) to get (1/2)^2 = 1/4.Multiply Results: Multiply the results from Step 3 and Step 5. Now we multiply 27 (from Step 3) by 1/4 (from Step 5) to get the final value of the expression. 27 * (1/4) = 27/4 = 6.75.

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((1)/(9))^(-(3)/(2))*((1)/(8))^((2)/(3))
What is the value of the given expression?

$\left(\frac{1}{9}\right)^{-\frac{3}{2}} \cdot\left(\frac{1}{8}\right)^{\frac{2}{3}}$ $\newline$ What is the value of the given expression?

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Algebra 1

Compare linear and exponential growth

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\left(\frac{1}{9}\right)^{-\frac{3}{2}} \cdot\left(\frac{1}{8}\right)^{\frac{2}{3}}

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What is the value of the given expression?

Understand and Simplify Exponents: Understand the given expression and simplify the exponents. $\newline$ The given expression is $\left(\frac{1}{9}\right)^{-\frac{3}{2}}\times\left(\frac{1}{8}\right)^{\frac{2}{3}}$ . We need to simplify each part of the expression separately.
Simplify $((1)/(9))^(-(3)/(2))$ : Simplify the first part of the expression $((1)/(9))^(-(3)/(2))$ . The negative exponent means we take the reciprocal of the base and then apply the positive exponent. So, $((1)/(9))^(-(3)/(2))$ becomes $(9)^{(3/2)}$ .
Calculate $(9)^{(3/2)}$ : Calculate $(9)^{(3/2)}$ . $\newline$ Since $9$ is a perfect square $(3^2)$ , we can rewrite $(9)^{(3/2)}$ as $(3^2)^{(3/2)}$ . When we raise a power to a power, we multiply the exponents, so $(3^2)^{(3/2)} = 3^{(2*(3/2))} = 3^3$ . $\newline$ Now calculate $3^3 = 3 \times 3 \times 3 = 27$ .
Simplify $((1)/(8))^((2)/(3))$ : Simplify the second part of the expression $((1)/(8))^((2)/(3))$ . The exponent $(2/3)$ means we take the cube root of the base and then square it. So, $((1)/(8))^((2)/(3))$ becomes $(1/2)^2$ .
Calculate $(1/2)^2$ : Calculate $(1/2)^2$ . $\newline$ Square the fraction $(1/2)$ to get $(1/2)^2 = 1/4$ .
Multiply Results: Multiply the results from Step $3$ and Step $5$ . $\newline$ Now we multiply $27$ (from Step $3$ ) by $\frac{1}{4}$ (from Step $5$ ) to get the final value of the expression. $\newline$ $27 \times \left(\frac{1}{4}\right) = \frac{27}{4} = 6.75$ .