Trigonometry Examples

$\sqrt[6]{x^{4} y^{2}} \sqrt[12]{x^{4} y^{44}}$

Step 1

Rewrite $x^{4} y^{2}$ as ${(x^{2} y)}^{2}$ .

$\sqrt[6]{{(x^{2} y)}^{2}} \sqrt[12]{x^{4} y^{44}}$

Step 2

Rewrite $\sqrt[6]{{(x^{2} y)}^{2}}$ as $\sqrt[3]{\sqrt{{(x^{2} y)}^{2}}}$ .

$\sqrt[3]{\sqrt{{(x^{2} y)}^{2}}} \sqrt[12]{x^{4} y^{44}}$

Step 3

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt[3]{x^{2} y} \sqrt[12]{x^{4} y^{44}}$

Step 4

Rewrite $x^{4} y^{44}$ as ${(y^{3})}^{12} (x^{4} y^{8})$ .

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Step 4.1

Factor out $y^{36}$ .

$\sqrt[3]{x^{2} y} \sqrt[12]{x^{4} (y^{36} y^{8})}$

Step 4.2

Rewrite $y^{36}$ as ${(y^{3})}^{12}$ .

$\sqrt[3]{x^{2} y} \sqrt[12]{x^{4} ({(y^{3})}^{12} y^{8})}$

Step 4.3

Reorder $x^{4}$ and ${(y^{3})}^{12}$ .

$\sqrt[3]{x^{2} y} \sqrt[12]{{(y^{3})}^{12} x^{4} y^{8}}$

Step 4.4

Add parentheses.

$\sqrt[3]{x^{2} y} \sqrt[12]{{(y^{3})}^{12} (x^{4} y^{8})}$

Step 5

Pull terms out from under the radical.

$\sqrt[3]{x^{2} y} (y^{3} \sqrt[12]{x^{4} y^{8}})$

Step 6

Rewrite $x^{4} y^{8}$ as ${(x y^{2})}^{4}$ .

$\sqrt[3]{x^{2} y} (y^{3} \sqrt[12]{{(x y^{2})}^{4}})$

Step 7

Rewrite $\sqrt[12]{{(x y^{2})}^{4}}$ as $\sqrt[3]{\sqrt[4]{{(x y^{2})}^{4}}}$ .

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

Step 8

Pull terms out from under the radical, assuming positive real numbers.

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

Step 9

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

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Step 9.1

Combine using the product rule for radicals.

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

Step 9.2

Raise $x$ to the power of $1$ .

$y^{3} \sqrt[3]{x^{1} x^{2} y \cdot y^{2}}$

Step 9.3

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

$y^{3} \sqrt[3]{x^{1 + 2} y \cdot y^{2}}$

Step 9.4

Add $1$ and $2$ .

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

Step 9.5

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 9.5.1

Move $y^{2}$ .

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

Step 9.5.2

Multiply $y^{2}$ by $y$ .

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Step 9.5.2.1

Raise $y$ to the power of $1$ .

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

Step 9.5.2.2

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

$y^{3} \sqrt[3]{x^{3} y^{2 + 1}}$

Step 9.5.3

Add $2$ and $1$ .

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

Step 10

Rewrite $x^{3} y^{3}$ as ${(x y)}^{3}$ .

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

Step 11

Pull terms out from under the radical, assuming real numbers.

$y^{3} (x y)$

Step 12

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 12.1

Move $y$ .

$y \cdot y^{3} x$

Step 12.2

Multiply $y$ by $y^{3}$ .

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Step 12.2.1

Raise $y$ to the power of $1$ .

$y^{1} y^{3} x$

Step 12.2.2

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

$y^{1 + 3} x$

Step 12.3

Add $1$ and $3$ .

$y^{4} x$