Finite Math Examples

$\sqrt[3]{x^{2} y^{3}} \sqrt[4]{x^{5} y^{4}}$

Step 1

Reorder $x^{2}$ and $y^{3}$ .

$\sqrt[3]{y^{3} x^{2}} \sqrt[4]{x^{5} y^{4}}$

Step 2

Pull terms out from under the radical.

$y \sqrt[3]{x^{2}} \sqrt[4]{x^{5} y^{4}}$

Step 3

Rewrite $x^{5} y^{4}$ as ${(x y)}^{4} x$ .

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Step 3.1

Factor out $x^{4}$ .

$y \sqrt[3]{x^{2}} \sqrt[4]{x^{4} x y^{4}}$

Step 3.2

Move $x$ .

$y \sqrt[3]{x^{2}} \sqrt[4]{x^{4} y^{4} x}$

Step 3.3

Rewrite $x^{4} y^{4}$ as ${(x y)}^{4}$ .

$y \sqrt[3]{x^{2}} \sqrt[4]{{(x y)}^{4} x}$

Step 4

Pull terms out from under the radical.

$y \sqrt[3]{x^{2}} (x y \sqrt[4]{x})$

Step 5

Multiply $y$ by $y$ by adding the exponents.

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Step 5.1

Move $y$ .

$y \cdot y \sqrt[3]{x^{2}} (x \sqrt[4]{x})$

Step 5.2

Multiply $y$ by $y$ .

$y^{2} \sqrt[3]{x^{2}} (x \sqrt[4]{x})$

Step 6

Multiply $y^{2} \sqrt[3]{x^{2}} (x \sqrt[4]{x})$ .

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Step 6.1

Rewrite the expression using the least common index of $12$ .

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Step 6.1.1

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

$y^{2} (x (x^{\frac{2}{3}} \sqrt[4]{x}))$

Step 6.1.2

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

$y^{2} (x (x^{\frac{8}{12}} \sqrt[4]{x}))$

Step 6.1.3

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

$y^{2} (x (\sqrt[12]{x^{8}} \sqrt[4]{x}))$

Step 6.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$y^{2} (x (\sqrt[12]{x^{8}} x^{\frac{1}{4}}))$

Step 6.1.5

Rewrite $x^{\frac{1}{4}}$ as $x^{\frac{3}{12}}$ .

$y^{2} (x (\sqrt[12]{x^{8}} x^{\frac{3}{12}}))$

Step 6.1.6

Rewrite $x^{\frac{3}{12}}$ as $\sqrt[12]{x^{3}}$ .

$y^{2} (x (\sqrt[12]{x^{8}} \sqrt[12]{x^{3}}))$

Step 6.2

Combine using the product rule for radicals.

$y^{2} (x \sqrt[12]{x^{8} x^{3}})$

Step 6.3

Multiply $x^{8}$ by $x^{3}$ by adding the exponents.

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Step 6.3.1

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

$y^{2} (x \sqrt[12]{x^{8 + 3}})$

Step 6.3.2

Add $8$ and $3$ .

$y^{2} (x \sqrt[12]{x^{11}})$

$y^{2} x \sqrt[12]{x^{11}}$