Pre-Algebra Examples

$\sqrt[5]{x^{16}} \sqrt[25]{x^{20}}$

Step 1

Rewrite $x^{16}$ as ${(x^{3})}^{5} x$ .

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Step 1.1

Factor out $x^{15}$ .

$\sqrt[5]{x^{15} x} \sqrt[25]{x^{20}}$

Step 1.2

Rewrite $x^{15}$ as ${(x^{3})}^{5}$ .

$\sqrt[5]{{(x^{3})}^{5} x} \sqrt[25]{x^{20}}$

Step 2

Pull terms out from under the radical.

$x^{3} \sqrt[5]{x} \sqrt[25]{x^{20}}$

Step 3

Rewrite $x^{20}$ as ${(x^{4})}^{5}$ .

$x^{3} \sqrt[5]{x} \sqrt[25]{{(x^{4})}^{5}}$

Step 4

Rewrite $\sqrt[25]{{(x^{4})}^{5}}$ as $\sqrt[5]{\sqrt[5]{{(x^{4})}^{5}}}$ .

$x^{3} \sqrt[5]{x} \sqrt[5]{\sqrt[5]{{(x^{4})}^{5}}}$

Step 5

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

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

Step 6

Multiply $x^{3} \sqrt[5]{x} \sqrt[5]{x^{4}}$ .

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 6.2.1

Multiply $x^{4}$ by $x$ .

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Step 6.2.1.1

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

$x^{3} \sqrt[5]{x^{4} x^{1}}$

Step 6.2.1.2

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

$x^{3} \sqrt[5]{x^{4 + 1}}$

Step 6.2.2

Add $4$ and $1$ .

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

Step 7

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

$x^{3} x$

Step 8

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

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

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

$x^{3 + 1}$

Step 8.2

Add $3$ and $1$ .

$x^{4}$