Solve for x square root of (x-1)^3=8

$\sqrt{{(x - 1)}^{3}} = 8$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{{(x - 1)}^{3}}}^{2} = 8^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(x - 1)}^{3}}$ as ${(x - 1)}^{\frac{3}{2}}$ .

${({(x - 1)}^{\frac{3}{2}})}^{2} = 8^{2}$

Simplify the left side.

Multiply the exponents in ${({(x - 1)}^{\frac{3}{2}})}^{2}$ .

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

${(x - 1)}^{\frac{3}{2} \cdot 2} = 8^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 1)}^{\frac{3}{2} \cdot 2} = 8^{2}$

Rewrite the expression.

${(x - 1)}^{3} = 8^{2}$

Simplify the right side.

Raise $8$ to the power of $2$ .

${(x - 1)}^{3} = 64$

Step 3

Solve for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 1 = \sqrt[3]{64}$

Simplify $\sqrt[3]{64}$ .

Rewrite $64$ as $4^{3}$ .

$x - 1 = \sqrt[3]{4^{3}}$

Pull terms out from under the radical, assuming real numbers.

$x - 1 = 4$

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

$x = 4 + 1$

Add $4$ and $1$ .

$x = 5$