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What could be the value of $x$ in the following equation? Select all that apply. $\newline$ $x^3 = \frac{1}{27}$ $\newline$ Multi-select Choices: $\newline$ (A) $\sqrt[3]{\frac{1}{27}}$ $\newline$ (B) $\frac{1}{3}$ $\newline$ (C) $-\sqrt[3]{\frac{1}{27}}$ $\newline$ (D) $-\frac{1}{3}$

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Evaluate integers raised to rational exponents

Full solution

Q. What could be the value of

x

in the following equation? Select all that apply.

\newline

x^3 = \frac{1}{27}

\newline

Multi-select Choices:

\newline

(A)

\sqrt[3]{\frac{1}{27}}

\newline

(B)

\frac{1}{3}

\newline

(C)

-\sqrt[3]{\frac{1}{27}}

\newline

(D)

-\frac{1}{3}

Identify Equation & Goal: Identify the equation and the goal. $\newline$ We need to find $x$ such that $x^3 = \frac{1}{27}$ .
Simplify Right Side: Simplify the right side of the equation. $\frac{1}{27}$ can be written as $(\frac{1}{3})^3$ .
Set Up Simplified Equation: Set up the equation with the simplified form. $x^3 = (\frac{1}{3})^3$
Solve for x: Solve for x by taking the cube root of both sides. $\newline$ $x = \sqrt[3]{(\frac{1}{3})^3}$ $\newline$ $x = \frac{1}{3}$
Check Other Solutions: Check for other possible solutions. $\newline$ Since the cube root function can also yield negative results for negative inputs, consider if $x^3 = -\frac{1}{27}$ could be a solution.
Simplify Negative Case: Simplify $-\frac{1}{27}$ as a cube. $\newline$ $-\frac{1}{27} = -\left(\frac{1}{27}\right) = -\left(\frac{1}{3}\right)^3$
Solve for x in Negative Case: Solve for x in the negative case. $\newline$ $x = \sqrt[3]{-(\frac{1}{3})^3}$ $\newline$ $x = -\frac{1}{3}$

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