Algebra Examples

$\sqrt[3]{18 x^{2} - 81 x} = x$

Step 1

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

${\sqrt[3]{18 x^{2} - 81 x}}^{3} = x^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{18 x^{2} - 81 x}$ as ${(18 x^{2} - 81 x)}^{\frac{1}{3}}$ .

${({(18 x^{2} - 81 x)}^{\frac{1}{3}})}^{3} = x^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(18 x^{2} - 81 x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(18 x^{2} - 81 x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

${(18 x^{2} - 81 x)}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(18 x^{2} - 81 x)}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(18 x^{2} - 81 x)}^{1} = x^{3}$

Step 2.2.1.2

Simplify.

$18 x^{2} - 81 x = x^{3}$

Step 3

Solve for $x$ .

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Step 3.1

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

$18 x^{2} - 81 x - x^{3} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- x$ out of $18 x^{2} - 81 x - x^{3}$ .

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Step 3.2.1.1

Reorder the expression.

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Step 3.2.1.1.1

Move $- 81 x$ .

$18 x^{2} - x^{3} - 81 x = 0$

Step 3.2.1.1.2

Reorder $18 x^{2}$ and $- x^{3}$ .

$- x^{3} + 18 x^{2} - 81 x = 0$

Step 3.2.1.2

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

$- x \cdot x^{2} + 18 x^{2} - 81 x = 0$

Step 3.2.1.3

Factor $- x$ out of $18 x^{2}$ .

$- x \cdot x^{2} - x (- 18 x) - 81 x = 0$

Step 3.2.1.4

Factor $- x$ out of $- 81 x$ .

$- x \cdot x^{2} - x (- 18 x) - x \cdot 81 = 0$

Step 3.2.1.5

Factor $- x$ out of $- x (x^{2}) - x (- 18 x)$ .

$- x (x^{2} - 18 x) - x \cdot 81 = 0$

Step 3.2.1.6

Factor $- x$ out of $- x (x^{2} - 18 x) - x (81)$ .

$- x (x^{2} - 18 x + 81) = 0$

Step 3.2.2

Factor using the perfect square rule.

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Step 3.2.2.1

Rewrite $81$ as $9^{2}$ .

$- x (x^{2} - 18 x + 9^{2}) = 0$

Step 3.2.2.2

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

$18 x = 2 \cdot x \cdot 9$

Step 3.2.2.3

Rewrite the polynomial.

$- x (x^{2} - 2 \cdot x \cdot 9 + 9^{2}) = 0$

Step 3.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 9$ .

$- x {(x - 9)}^{2} = 0$

Step 3.3

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

$x = 0$

${(x - 9)}^{2} = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set ${(x - 9)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set ${(x - 9)}^{2}$ equal to $0$ .

${(x - 9)}^{2} = 0$

Step 3.5.2

Solve ${(x - 9)}^{2} = 0$ for $x$ .

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Step 3.5.2.1

Set the $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 3.5.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 3.6

The final solution is all the values that make $- x {(x - 9)}^{2} = 0$ true.

$x = 0, 9$