Algebra Examples

$\sqrt[3]{128} - \sqrt[3]{16} = a \sqrt[3]{b}$

Step 1

Rewrite the equation as $a \sqrt[3]{b} = \sqrt[3]{128} - \sqrt[3]{16}$ .

$a \sqrt[3]{b} = \sqrt[3]{128} - \sqrt[3]{16}$

Step 2

Simplify.

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Step 2.1

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 2.1.1

Factor $8$ out of $16$ .

$a \sqrt[3]{b} = \sqrt[3]{128} - \sqrt[3]{8 (2)}$

Step 2.1.2

Rewrite $8$ as $2^{3}$ .

$a \sqrt[3]{b} = \sqrt[3]{128} - \sqrt[3]{2^{3} \cdot 2}$

Step 2.2

Pull terms out from under the radical.

$a \sqrt[3]{b} = \sqrt[3]{128} - (2 \sqrt[3]{2})$

Step 2.3

Multiply $2$ by $- 1$ .

$a \sqrt[3]{b} = \sqrt[3]{128} - 2 \sqrt[3]{2}$

Step 3

Divide each term in $a \sqrt[3]{b} = \sqrt[3]{128} - 2 \sqrt[3]{2}$ by $\sqrt[3]{b}$ and simplify.

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Step 3.1

Divide each term in $a \sqrt[3]{b} = \sqrt[3]{128} - 2 \sqrt[3]{2}$ by $\sqrt[3]{b}$ .

$\frac{a \sqrt[3]{b}}{\sqrt[3]{b}} = \frac{\sqrt[3]{128}}{\sqrt[3]{b}} + \frac{- 2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\sqrt[3]{b}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{a \sqrt[3]{b}}{\sqrt[3]{b}} = \frac{\sqrt[3]{128}}{\sqrt[3]{b}} + \frac{- 2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.2.1.2

Divide $a$ by $1$ .

$a = \frac{\sqrt[3]{128}}{\sqrt[3]{b}} + \frac{- 2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine the numerators over the common denominator.

$a = \frac{\sqrt[3]{128} - 2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Rewrite $128$ as $4^{3} \cdot 2$ .

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Step 3.3.2.1.1

Factor $64$ out of $128$ .

$a = \frac{\sqrt[3]{64 (2)} - 2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.3.2.1.2

Rewrite $64$ as $4^{3}$ .

$a = \frac{\sqrt[3]{4^{3} \cdot 2} - 2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.3.2.2

Pull terms out from under the radical.

$a = \frac{4 \sqrt[3]{2} - 2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.3.3

Subtract $2 \sqrt[3]{2}$ from $4 \sqrt[3]{2}$ .

$a = \frac{2 \sqrt[3]{2}}{\sqrt[3]{b}}$

Step 3.3.4

Multiply $\frac{2 \sqrt[3]{2}}{\sqrt[3]{b}}$ by $\frac{{\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{2}}$ .

$a = \frac{2 \sqrt[3]{2}}{\sqrt[3]{b}} \cdot \frac{{\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{2}}$

Step 3.3.5

Combine and simplify the denominator.

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Step 3.3.5.1

Multiply $\frac{2 \sqrt[3]{2}}{\sqrt[3]{b}}$ by $\frac{{\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{2}}$ .

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{\sqrt[3]{b} {\sqrt[3]{b}}^{2}}$

Step 3.3.5.2

Raise $\sqrt[3]{b}$ to the power of $1$ .

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{1} {\sqrt[3]{b}}^{2}}$

Step 3.3.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{1 + 2}}$

Step 3.3.5.4

Add $1$ and $2$ .

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{3}}$

Step 3.3.5.5

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

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Step 3.3.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{b}$ as $b^{\frac{1}{3}}$ .

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

Step 3.3.5.5.2

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

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{\frac{1}{3} \cdot 3}}$

Step 3.3.5.5.3

Combine $\frac{1}{3}$ and $3$ .

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{\frac{3}{3}}}$

Step 3.3.5.5.4

Cancel the common factor of $3$ .

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Step 3.3.5.5.4.1

Cancel the common factor.

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{\frac{3}{3}}}$

Step 3.3.5.5.4.2

Rewrite the expression.

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{1}}$

Step 3.3.5.5.5

Simplify.

$a = \frac{2 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b}$

Step 3.3.6

Simplify the numerator.

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Step 3.3.6.1

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

$a = \frac{2 \sqrt[3]{2} \sqrt[3]{b^{2}}}{b}$

Step 3.3.6.2

Combine using the product rule for radicals.

$a = \frac{2 \sqrt[3]{b^{2} \cdot 2}}{b}$

Step 3.3.7

Move $2$ to the left of $b^{2}$ .

$a = \frac{2 \sqrt[3]{2 b^{2}}}{b}$