Precalculus Examples

$\sqrt[5]{y^{11}} \sqrt[25]{y^{20}}$

Step 1

Rewrite $y^{11}$ as ${(y^{2})}^{5} y$ .

Tap for more steps...

Step 1.1

Factor out $y^{10}$ .

$\sqrt[5]{y^{10} y} \sqrt[25]{y^{20}}$

Step 1.2

Rewrite $y^{10}$ as ${(y^{2})}^{5}$ .

$\sqrt[5]{{(y^{2})}^{5} y} \sqrt[25]{y^{20}}$

Step 2

Pull terms out from under the radical.

$y^{2} \sqrt[5]{y} \sqrt[25]{y^{20}}$

Step 3

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

$y^{2} \sqrt[5]{y} \sqrt[25]{{(y^{4})}^{5}}$

Step 4

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

$y^{2} \sqrt[5]{y} \sqrt[5]{\sqrt[5]{{(y^{4})}^{5}}}$

Step 5

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

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

Step 6

Multiply $y^{2} \sqrt[5]{y} \sqrt[5]{y^{4}}$ .

Tap for more steps...

Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

Tap for more steps...

Step 6.2.1.1

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

$y^{2} \sqrt[5]{y^{4} y^{1}}$

Step 6.2.1.2

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

$y^{2} \sqrt[5]{y^{4 + 1}}$

Step 6.2.2

Add $4$ and $1$ .

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

Step 7

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

$y^{2} y$

Step 8

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

Tap for more steps...

Step 8.1

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

Tap for more steps...

Step 8.1.1

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

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

Step 8.1.2

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

$y^{2 + 1}$

Step 8.2

Add $2$ and $1$ .

$y^{3}$