Rephrase the Problem: First, let's rephrase the ＂What is the value of x if x raised to the power of negative three-halves equals one over seven hundred twenty-nine?＂Identify Equation and Value: Identify the given equation and the value we need to find. We have the equation x^(-3/2) = 1/729, and we need to find the value of x.Recognize Perfect Cube: Recognize that 729 is a perfect cube, as 729 = 9^3.Rewrite Equation: Since x^(-3/2) = 1/729, we can rewrite 1/729 as (1/9)^3 to match the exponent on x. So, x^(-3/2) = (1/9)^3.Take Reciprocal: Now, we can take the reciprocal of both sides to get rid of the negative exponent. This gives us x^(3/2) = 9^3.Cancel Exponents: To find x, we need to take both sides to the power of 2/3 to cancel out the exponent of 3/2 on x. (x^(3/2))^(2/3) = (9^3)^(2/3).Simplify Exponents: When we raise a power to a power, we multiply the exponents. So, x^(3/2 * 2/3) = 9^(3 * 2/3).Calculate Value: Simplify the exponents. x^(1) = 9^2.Final Solution: Calculate 9^2. 9^2 = 81.Now we have x = 81, which is the solution to the original equation.

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$x^{-\frac{3}{2}} = \frac{1}{729}$

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Algebra 2

Add, subtract, multiply, and divide polynomials

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x^{-\frac{3}{2}} = \frac{1}{729}

Rephrase the Problem: First, let's rephrase the ＂What is the value of $x$ if $x$ raised to the power of negative three-halves equals one over seven hundred twenty-nine?＂
Identify Equation and Value: Identify the given equation and the value we need to find. $\newline$ We have the equation $x^{(-3/2)} = \frac{1}{729}$ , and we need to find the value of $x$ .
Recognize Perfect Cube: Recognize that $729$ is a perfect cube, as $729 = 9^3$ .
Rewrite Equation: Since $x^{-\frac{3}{2}} = \frac{1}{729}$ , we can rewrite $\frac{1}{729}$ as $(\frac{1}{9})^3$ to match the exponent on $x$ . So, $x^{-\frac{3}{2}} = (\frac{1}{9})^3$ .
Take Reciprocal: Now, we can take the reciprocal of both sides to get rid of the negative exponent. $\newline$ This gives us $x^{\frac{3}{2}} = 9^3$ .
Cancel Exponents: To find $x$ , we need to take both sides to the power of $\frac{2}{3}$ to cancel out the exponent of $\frac{3}{2}$ on $x$ . $(x^{\frac{3}{2}})^{\frac{2}{3}} = (9^3)^{\frac{2}{3}}$ .
Simplify Exponents: When we raise a power to a power, we multiply the exponents. $\newline$ So, $x^{(3/2) \times (2/3)} = 9^{(3 \times 2/3)}$ .
Calculate Value: Simplify the exponents. $x^{(1)} = 9^{2}$ .
Final Solution: Calculate $9^2$ . $\newline$ $9^2 = 81$ .
Final Solution: Calculate $9^2$ . $9^2 = 81$ . Now we have $x = 81$ , which is the solution to the original equation.