Solve for x cube root of 16x^2-64x=x

$\sqrt[3]{16 x^{2} - 64 x} = x$

Step 1

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

${\sqrt[3]{16 x^{2} - 64 x}}^{3} = x^{3}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{16 x^{2} - 64 x}$ as ${(16 x^{2} - 64 x)}^{\frac{1}{3}}$ .

${({(16 x^{2} - 64 x)}^{\frac{1}{3}})}^{3} = x^{3}$

Simplify the left side.

Simplify ${({(16 x^{2} - 64 x)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(16 x^{2} - 64 x)}^{\frac{1}{3}})}^{3}$ .

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

${(16 x^{2} - 64 x)}^{\frac{1}{3} \cdot 3} = x^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(16 x^{2} - 64 x)}^{\frac{1}{3} \cdot 3} = x^{3}$

Rewrite the expression.

${(16 x^{2} - 64 x)}^{1} = x^{3}$

Simplify.

$16 x^{2} - 64 x = x^{3}$

Step 3

Solve for $x$ .

Subtract $x^{3}$ from both sides of the equation.

$16 x^{2} - 64 x - x^{3} = 0$

Factor the left side of the equation.

Factor $- x$ out of $16 x^{2} - 64 x - x^{3}$ .

Reorder the expression.

Move $- 64 x$ .

$16 x^{2} - x^{3} - 64 x = 0$

Reorder $16 x^{2}$ and $- x^{3}$ .

$- x^{3} + 16 x^{2} - 64 x = 0$

Factor $- x$ out of $- x^{3}$ .

$- x \cdot x^{2} + 16 x^{2} - 64 x = 0$

Factor $- x$ out of $16 x^{2}$ .

$- x \cdot x^{2} - x (- 16 x) - 64 x = 0$

Factor $- x$ out of $- 64 x$ .

$- x \cdot x^{2} - x (- 16 x) - x \cdot 64 = 0$

Factor $- x$ out of $- x (x^{2}) - x (- 16 x)$ .

$- x (x^{2} - 16 x) - x \cdot 64 = 0$

Factor $- x$ out of $- x (x^{2} - 16 x) - x (64)$ .

$- x (x^{2} - 16 x + 64) = 0$

Factor using the perfect square rule.

Rewrite $64$ as $8^{2}$ .

$- x (x^{2} - 16 x + 8^{2}) = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 x = 2 \cdot x \cdot 8$

Rewrite the polynomial.

$- x (x^{2} - 2 \cdot x \cdot 8 + 8^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 8$ .

$- x {(x - 8)}^{2} = 0$

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

$x = 0$

${(x - 8)}^{2} = 0$

Set $x$ equal to $0$ .

$x = 0$

Set ${(x - 8)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 8)}^{2}$ equal to $0$ .

${(x - 8)}^{2} = 0$

Solve ${(x - 8)}^{2} = 0$ for $x$ .

Set the $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

The final solution is all the values that make $- x {(x - 8)}^{2} = 0$ true.

$x = 0, 8$

Solve for x cube root of 16x^2-64x=x

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