Solve for x fourth root of x^3=64

$\sqrt[4]{x^{3}} = 64$

Step 1

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

${\sqrt[4]{x^{3}}}^{4} = 64^{4}$

Step 2

Simplify each side of the equation.

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

${(x^{\frac{3}{4}})}^{4} = 64^{4}$

Simplify the left side.

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

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

$x^{\frac{3}{4} \cdot 4} = 64^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x^{\frac{3}{4} \cdot 4} = 64^{4}$

Rewrite the expression.

$x^{3} = 64^{4}$

Simplify the right side.

Raise $64$ to the power of $4$ .

$x^{3} = 16777216$

Step 3

Solve for $x$ .

Subtract $16777216$ from both sides of the equation.

$x^{3} - 16777216 = 0$

Factor the left side of the equation.

Rewrite $16777216$ as $256^{3}$ .

$x^{3} - 256^{3} = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 256$ .

$(x - 256) (x^{2} + x \cdot 256 + 256^{2}) = 0$

Simplify.

Move $256$ to the left of $x$ .

$(x - 256) (x^{2} + 256 x + 256^{2}) = 0$

Raise $256$ to the power of $2$ .

$(x - 256) (x^{2} + 256 x + 65536) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 256 = 0$

$x^{2} + 256 x + 65536 = 0$

Set $x - 256$ equal to $0$ and solve for $x$ .

Set $x - 256$ equal to $0$ .

$x - 256 = 0$

Add $256$ to both sides of the equation.

$x = 256$

Set $x^{2} + 256 x + 65536$ equal to $0$ and solve for $x$ .

Set $x^{2} + 256 x + 65536$ equal to $0$ .

$x^{2} + 256 x + 65536 = 0$

Solve $x^{2} + 256 x + 65536 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 256$ , and $c = 65536$ into the quadratic formula and solve for $x$ .

$\frac{- 256 \pm \sqrt{256^{2} - 4 \cdot (1 \cdot 65536)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $256$ to the power of $2$ .

$x = \frac{- 256 \pm \sqrt{65536 - 4 \cdot 1 \cdot 65536}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 65536$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 256 \pm \sqrt{65536 - 4 \cdot 65536}}{2 \cdot 1}$

Multiply $- 4$ by $65536$ .

$x = \frac{- 256 \pm \sqrt{65536 - 262144}}{2 \cdot 1}$

Subtract $262144$ from $65536$ .

$x = \frac{- 256 \pm \sqrt{- 196608}}{2 \cdot 1}$

Rewrite $- 196608$ as $- 1 (196608)$ .

$x = \frac{- 256 \pm \sqrt{- 1 \cdot 196608}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (196608)}$ as $\sqrt{- 1} \cdot \sqrt{196608}$ .

$x = \frac{- 256 \pm \sqrt{- 1} \cdot \sqrt{196608}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 256 \pm i \cdot \sqrt{196608}}{2 \cdot 1}$

Rewrite $196608$ as $256^{2} \cdot 3$ .

Factor $65536$ out of $196608$ .

$x = \frac{- 256 \pm i \cdot \sqrt{65536 (3)}}{2 \cdot 1}$

Rewrite $65536$ as $256^{2}$ .

$x = \frac{- 256 \pm i \cdot \sqrt{256^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 256 \pm i \cdot (256 \sqrt{3})}{2 \cdot 1}$

Move $256$ to the left of $i$ .

$x = \frac{- 256 \pm 256 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 256 \pm 256 i \sqrt{3}}{2}$

Simplify $\frac{- 256 \pm 256 i \sqrt{3}}{2}$ .

$x = - 128 \pm 128 i \sqrt{3}$

The final answer is the combination of both solutions.

$x = - 128 + 128 i \sqrt{3}, - 128 - 128 i \sqrt{3}$

The final solution is all the values that make $(x - 256) (x^{2} + 256 x + 65536) = 0$ true.

$x = 256, - 128 + 128 i \sqrt{3}, - 128 - 128 i \sqrt{3}$

Solve for x fourth root of x^3=64

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