Question
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Tutor-Verified Answer
Answer
The solutions are \(x=-27\) and \(x=27\).
Solution
Solve the equation \( x^{\frac{2}{3}}=9 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(x^{\frac{2}{3}}=9\)
- step1: Simplify the expression:
\(x=\pm 9^{\frac{3}{2}}\)
- step2: Simplify:
\(x=\pm 27\)
- step3: Separate into possible cases:
\(\begin{align}&x=27\\&x=-27\end{align}\)
- step4: Rewrite:
\(x_{1}=-27,x_{2}=27\)
The solutions to the equation \(x^{\frac{2}{3}}=9\) are \(x=-27\) and \(x=27\).
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Simplify this solution The Deep Dive
To solve \( x^{\frac{2}{3}}=9 \), start by raising both sides to the power of \( \frac{3}{2} \) to isolate \( x \). This gives you: \[ x = 9^{\frac{3}{2}}. \] Calculate the right side by first finding the square root of 9, which is 3, and then cubing that result: \( 3^3 = 27 \). Therefore, \( x = 27 \). Now, to check your answer, plug \( x = 27 \) back into the original equation: \[ (27)^{\frac{2}{3}} = 9. \] This is indeed true, confirming that \( x = 27 \) is the correct solution!
