Finite Math Examples

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

Step 1

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

${(2 \sqrt[3]{x})}^{3} = {\sqrt[3]{x^{2}}}^{3}$

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

${(2 x^{\frac{1}{3}})}^{3} = {\sqrt[3]{x^{2}}}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the product rule to $2 x^{\frac{1}{3}}$ .

$2^{3} {(x^{\frac{1}{3}})}^{3} = {\sqrt[3]{x^{2}}}^{3}$

Step 2.2.1.2

Raise $2$ to the power of $3$ .

$8 {(x^{\frac{1}{3}})}^{3} = {\sqrt[3]{x^{2}}}^{3}$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$8 x^{\frac{1}{3} \cdot 3} = {\sqrt[3]{x^{2}}}^{3}$

Step 2.2.1.3.2

Cancel the common factor of $3$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$8 x^{\frac{1}{3} \cdot 3} = {\sqrt[3]{x^{2}}}^{3}$

Step 2.2.1.3.2.2

Rewrite the expression.

$8 x^{1} = {\sqrt[3]{x^{2}}}^{3}$

Step 2.2.1.4

Simplify.

$8 x = {\sqrt[3]{x^{2}}}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt[3]{x^{2}}}^{3}$ as $x^{2}$ .

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

$8 x = x^{\frac{2}{3} \cdot 3}$

Step 2.3.1.3

Combine $\frac{2}{3}$ and $3$ .

$8 x = x^{\frac{2 \cdot 3}{3}}$

Step 2.3.1.4

Multiply $2$ by $3$ .

$8 x = x^{\frac{6}{3}}$

Step 2.3.1.5

Cancel the common factor of $6$ and $3$ .

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Step 2.3.1.5.1

Factor $3$ out of $6$ .

$8 x = x^{\frac{3 \cdot 2}{3}}$

Step 2.3.1.5.2

Cancel the common factors.

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Step 2.3.1.5.2.1

Factor $3$ out of $3$ .

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

Step 2.3.1.5.2.2

Cancel the common factor.

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

Step 2.3.1.5.2.3

Rewrite the expression.

$8 x = x^{\frac{2}{1}}$

Step 2.3.1.5.2.4

Divide $2$ by $1$ .

$8 x = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$8 u - u^{2} = 0$

Step 3.2.2

Factor $u$ out of $8 u - u^{2}$ .

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Step 3.2.2.1

Factor $u$ out of $8 u$ .

$u \cdot 8 - u^{2} = 0$

Step 3.2.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 8 + u (- u) = 0$

Step 3.2.2.3

Factor $u$ out of $u \cdot 8 + u (- u)$ .

$u (8 - u) = 0$

Step 3.2.3

Replace all occurrences of $u$ with $x$ .

$x (8 - x) = 0$

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 $0$ .

$x = 0$

$8 - x = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $8 - x$ equal to $0$ and solve for $x$ .

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Step 3.5.1

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

$8 - x = 0$

Step 3.5.2

Solve $8 - x = 0$ for $x$ .

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Step 3.5.2.1

Subtract $8$ from both sides of the equation.

$- x = - 8$

Step 3.5.2.2

Divide each term in $- x = - 8$ by $- 1$ and simplify.

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Step 3.5.2.2.1

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Step 3.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Divide $- 8$ by $- 1$ .

$x = 8$

Step 3.6

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

$x = 0, 8$