Precalculus Examples

$\sqrt[5]{x^{3} y^{2}} \sqrt[10]{x^{4} y^{26}}$

Step 1

Rewrite $x^{4} y^{26}$ as ${(y^{2})}^{10} (x^{4} y^{6})$ .

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Step 1.1

Factor out $y^{20}$ .

$\sqrt[5]{x^{3} y^{2}} \sqrt[10]{x^{4} (y^{20} y^{6})}$

Step 1.2

Rewrite $y^{20}$ as ${(y^{2})}^{10}$ .

$\sqrt[5]{x^{3} y^{2}} \sqrt[10]{x^{4} ({(y^{2})}^{10} y^{6})}$

Step 1.3

Reorder $x^{4}$ and ${(y^{2})}^{10}$ .

$\sqrt[5]{x^{3} y^{2}} \sqrt[10]{{(y^{2})}^{10} x^{4} y^{6}}$

Step 1.4

Add parentheses.

$\sqrt[5]{x^{3} y^{2}} \sqrt[10]{{(y^{2})}^{10} (x^{4} y^{6})}$

Step 2

Pull terms out from under the radical.

$\sqrt[5]{x^{3} y^{2}} (y^{2} \sqrt[10]{x^{4} y^{6}})$

Step 3

Rewrite $x^{4} y^{6}$ as ${(x^{2} y^{3})}^{2}$ .

$\sqrt[5]{x^{3} y^{2}} (y^{2} \sqrt[10]{{(x^{2} y^{3})}^{2}})$

Step 4

Rewrite $\sqrt[10]{{(x^{2} y^{3})}^{2}}$ as $\sqrt[5]{\sqrt{{(x^{2} y^{3})}^{2}}}$ .

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

Step 5

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

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

Step 6

Multiply $\sqrt[5]{x^{3} y^{2}} (y^{2} \sqrt[5]{x^{2} y^{3}})$ .

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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

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

Step 6.3

Add $2$ and $3$ .

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

Step 6.4

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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Step 6.4.1

Move $y^{3}$ .

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

Step 6.4.2

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

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

Step 6.4.3

Add $3$ and $2$ .

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

Step 7

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

$y^{2} \sqrt[5]{{(x y)}^{5}}$

Step 8

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

$y^{2} (x y)$

Step 9

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

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Step 9.1

Move $y$ .

$y \cdot y^{2} x$

Step 9.2

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

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Step 9.2.1

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

$y^{1} y^{2} x$

Step 9.2.2

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

$y^{1 + 2} x$

Step 9.3

Add $1$ and $2$ .

$y^{3} x$