Solve for x fourth root of (2x-1)^3=27

$\sqrt[4]{{(2 x - 1)}^{3}} = 27$

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{{(2 x - 1)}^{3}}}^{4} = 27^{4}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{{(2 x - 1)}^{3}}$ as ${(2 x - 1)}^{\frac{3}{4}}$ .

${({(2 x - 1)}^{\frac{3}{4}})}^{4} = 27^{4}$

Simplify the left side.

Multiply the exponents in ${({(2 x - 1)}^{\frac{3}{4}})}^{4}$ .

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

${(2 x - 1)}^{\frac{3}{4} \cdot 4} = 27^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(2 x - 1)}^{\frac{3}{4} \cdot 4} = 27^{4}$

Rewrite the expression.

${(2 x - 1)}^{3} = 27^{4}$

Simplify the right side.

Raise $27$ to the power of $4$ .

${(2 x - 1)}^{3} = 531441$

Solve for $x$ .

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

$2 x - 1 = \sqrt[3]{531441}$

Simplify $\sqrt[3]{531441}$ .

Rewrite $531441$ as $81^{3}$ .

$2 x - 1 = \sqrt[3]{81^{3}}$

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

$2 x - 1 = 81$

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

Add $1$ to both sides of the equation.

$2 x = 81 + 1$

Add $81$ and $1$ .

$2 x = 82$

Divide each term in $2 x = 82$ by $2$ and simplify.

Divide each term in $2 x = 82$ by $2$ .

$\frac{2 x}{2} = \frac{82}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{82}{2}$

Divide $x$ by $1$ .

$x = \frac{82}{2}$

Simplify the right side.

Divide $82$ by $2$ .

$x = 41$