Basic Math Examples

$\sqrt{5 y^{5}} \cdot \sqrt{15 y}$

Step 1

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

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Step 1.1

Factor out $y^{4}$ .

$\sqrt{5 (y^{4} y)} \cdot \sqrt{15 y}$

Step 1.2

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

$\sqrt{5 ({(y^{2})}^{2} y)} \cdot \sqrt{15 y}$

Step 1.3

Reorder $5$ and ${(y^{2})}^{2}$ .

$\sqrt{{(y^{2})}^{2} \cdot 5 y} \cdot \sqrt{15 y}$

Step 1.4

Add parentheses.

$\sqrt{{(y^{2})}^{2} \cdot (5 y)} \cdot \sqrt{15 y}$

Step 2

Pull terms out from under the radical.

$y^{2} \sqrt{5 y} \cdot \sqrt{15 y}$

Step 3

Multiply $y^{2} \sqrt{5 y} \sqrt{15 y}$ .

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Step 3.1

Combine using the product rule for radicals.

$y^{2} \sqrt{15 y (5 y)}$

Step 3.2

Multiply $5$ by $15$ .

$y^{2} \sqrt{75 y \cdot y}$

Step 3.3

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

$y^{2} \sqrt{75 (y^{1} y)}$

Step 3.4

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

$y^{2} \sqrt{75 (y^{1} y^{1})}$

Step 3.5

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

$y^{2} \sqrt{75 y^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$y^{2} \sqrt{75 y^{2}}$

Step 4

Rewrite $75 y^{2}$ as ${(5 y)}^{2} \cdot 3$ .

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Step 4.1

Factor $25$ out of $75$ .

$y^{2} \sqrt{25 (3) y^{2}}$

Step 4.2

Rewrite $25$ as $5^{2}$ .

$y^{2} \sqrt{5^{2} \cdot 3 y^{2}}$

Step 4.3

Move $3$ .

$y^{2} \sqrt{5^{2} y^{2} \cdot 3}$

Step 4.4

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

$y^{2} \sqrt{{(5 y)}^{2} \cdot 3}$

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite using the commutative property of multiplication.

$5 \sqrt{3} y^{2} y$

Step 7

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

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Step 7.1

Move $y$ .

$5 \sqrt{3} (y \cdot y^{2})$

Step 7.2

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

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Step 7.2.1

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

$5 \sqrt{3} (y^{1} y^{2})$

Step 7.2.2

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

$5 \sqrt{3} y^{1 + 2}$

Step 7.3

Add $1$ and $2$ .

$5 \sqrt{3} y^{3}$