Questions: Solve for (x) in the equation. [x^10/3=27 cdot x^1/3]

Transcript text: Solve for $x$ in the equation. \[ x^{\frac{10}{3}}=27 \cdot x^{\frac{1}{3}} \]

Solution

Solution Steps

Step 1: Rewrite the equation

The given equation is: \[ x^{\frac{10}{3}} = 27 \cdot x^{\frac{1}{3}} \]

To simplify, divide both sides by $ x^{\frac{1}{3}} $ (assuming $ x \neq 0 $): \[ x^{\frac{10}{3}} \div x^{\frac{1}{3}} = 27 \]

Step 2: Simplify the exponents

Using the property of exponents $ x^a \div x^b = x^{a-b} $, we get: \[ x^{\frac{10}{3} - \frac{1}{3}} = 27 \] \[ x^{\frac{9}{3}} = 27 \] \[ x^3 = 27 \]

Step 3: Solve for $ x $

Take the cube root of both sides: \[ x = \sqrt[3]{27} \] \[ x = 3 \]

Final Answer

\[ \boxed{x = 3} \]

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