Solve by Factoring cube root of x^5=32

$\sqrt[3]{x^{5}} = 32$

Step 1

Subtract $32$ from both sides of the equation.

$\sqrt[3]{x^{5}} - 32 = 0$

Step 2

Simplify each term.

Factor out $x^{3}$ .

$\sqrt[3]{x^{3} x^{2}} - 32 = 0$

Pull terms out from under the radical.

$x \sqrt[3]{x^{2}} - 32 = 0$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$x \cdot x^{\frac{2}{3}} - 32 = 0$

Step 4

Move the terms containing $x$ to the left side and simplify.

Add $32$ to both sides of the equation.

$x \cdot x^{\frac{2}{3}} = 32$

Multiply $x$ by $x^{\frac{2}{3}}$ by adding the exponents.

Multiply $x$ by $x^{\frac{2}{3}}$ .

Raise $x$ to the power of $1$ .

$x \cdot x^{\frac{2}{3}} = 32$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + \frac{2}{3}} = 32$

Write $1$ as a fraction with a common denominator.

$x^{\frac{3}{3} + \frac{2}{3}} = 32$

Combine the numerators over the common denominator.

$x^{\frac{3 + 2}{3}} = 32$

Add $3$ and $2$ .

$x^{\frac{5}{3}} = 32$

Step 5

Raise each side of the equation to the power of $\frac{3}{5}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{5}{3}})}^{\frac{3}{5}} = 32^{\frac{3}{5}}$

Step 6

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{5}{3}})}^{\frac{3}{5}}$ .

Multiply the exponents in ${(x^{\frac{5}{3}})}^{\frac{3}{5}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{5}{3} \cdot \frac{3}{5}} = 32^{\frac{3}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{5}{3} \cdot \frac{3}{5}} = 32^{\frac{3}{5}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = 32^{\frac{3}{5}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 32^{\frac{3}{5}}$

Rewrite the expression.

$x^{1} = 32^{\frac{3}{5}}$

Simplify.

$x = 32^{\frac{3}{5}}$

Simplify the right side.

Simplify $32^{\frac{3}{5}}$ .

Simplify the expression.

Rewrite $32$ as $2^{5}$ .

$x = {(2^{5})}^{\frac{3}{5}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 2^{5 (\frac{3}{5})}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x = 2^{5 (\frac{3}{5})}$

Rewrite the expression.

$x = 2^{3}$

Raise $2$ to the power of $3$ .

$x = 8$

Solve by Factoring cube root of x^5=32

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