Algebra Examples

$\sqrt[3]{5 x y^{9}} + \sqrt[3]{5 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 x y^{9}}$ as ${(5 x y^{9})}^{\frac{1}{3}}$ .

${(5 x y^{9})}^{\frac{1}{3}} + \sqrt[3]{5 x}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 x}$ as ${(5 x)}^{\frac{1}{3}}$ .

${(5 x y^{9})}^{\frac{1}{3}} + {(5 x)}^{\frac{1}{3}}$

Step 3

Use the power rule ${(a i)}^{n} = a^{n} i^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $5 x y^{9}$ .

${(5 x)}^{\frac{1}{3}} {(y^{9})}^{\frac{1}{3}} + {(5 x)}^{\frac{1}{3}}$

Step 3.2

Apply the product rule to $5 x$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} {(y^{9})}^{\frac{1}{3}} + {(5 x)}^{\frac{1}{3}}$

Step 4

Multiply the exponents in ${(y^{9})}^{\frac{1}{3}}$ .

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Step 4.1

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

$5^{\frac{1}{3}} x^{\frac{1}{3}} y^{9 (\frac{1}{3})} + {(5 x)}^{\frac{1}{3}}$

Step 4.2

Cancel the common factor of $3$ .

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Step 4.2.1

Factor $3$ out of $9$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} y^{3 (3) \frac{1}{3}} + {(5 x)}^{\frac{1}{3}}$

Step 4.2.2

Cancel the common factor.

$5^{\frac{1}{3}} x^{\frac{1}{3}} y^{3 \cdot 3 \frac{1}{3}} + {(5 x)}^{\frac{1}{3}}$

Step 4.2.3

Rewrite the expression.

$5^{\frac{1}{3}} x^{\frac{1}{3}} y^{3} + {(5 x)}^{\frac{1}{3}}$

Step 5

Apply the product rule to $5 x$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} y^{3} + 5^{\frac{1}{3}} x^{\frac{1}{3}}$

Step 6

Rewrite $5^{\frac{1}{3}} x^{\frac{1}{3}} y^{3} + 5^{\frac{1}{3}} x^{\frac{1}{3}}$ in a factored form.

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Step 6.1

Factor $5^{\frac{1}{3}} x^{\frac{1}{3}}$ out of $5^{\frac{1}{3}} x^{\frac{1}{3}} y^{3} + 5^{\frac{1}{3}} x^{\frac{1}{3}}$ .

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Step 6.1.1

Factor $5^{\frac{1}{3}} x^{\frac{1}{3}}$ out of $5^{\frac{1}{3}} x^{\frac{1}{3}} y^{3}$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} (y^{3}) + 5^{\frac{1}{3}} x^{\frac{1}{3}}$

Step 6.1.2

Factor $5^{\frac{1}{3}} x^{\frac{1}{3}}$ out of $5^{\frac{1}{3}} x^{\frac{1}{3}}$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} (y^{3}) + 5^{\frac{1}{3}} x^{\frac{1}{3}} (1)$

Step 6.1.3

Factor $5^{\frac{1}{3}} x^{\frac{1}{3}}$ out of $5^{\frac{1}{3}} x^{\frac{1}{3}} (y^{3}) + 5^{\frac{1}{3}} x^{\frac{1}{3}} (1)$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} (y^{3} + 1)$

Step 6.2

Rewrite $1$ as $1^{3}$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} (y^{3} + 1^{3})$

Step 6.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + i^{3} = (a + i) (a^{2} - a i + i^{2})$ where $a = y$ and $i = 1$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} ((y + 1) (y^{2} - y \cdot 1 + 1^{2}))$

Step 6.4

Factor.

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Step 6.4.1

Simplify.

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Step 6.4.1.1

Multiply $- 1$ by $1$ .

$5^{\frac{1}{3}} x^{\frac{1}{3}} ((y + 1) (y^{2} - y + 1^{2}))$

Step 6.4.1.2

One to any power is one.

$5^{\frac{1}{3}} x^{\frac{1}{3}} ((y + 1) (y^{2} - y + 1))$

Step 6.4.2

Remove unnecessary parentheses.

$5^{\frac{1}{3}} x^{\frac{1}{3}} (y + 1) (y^{2} - y + 1)$