Basic Math Examples

$4 a y \sqrt{3 a^{5}} \cdot (3 y \sqrt{6 y^{4}})$

Step 1

Multiply $y$ by $y$ by adding the exponents.

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Step 1.1

Move $y$ .

$4 a (y \cdot y) \sqrt{3 a^{5}} \cdot (3 \sqrt{6 y^{4}})$

Step 1.2

Multiply $y$ by $y$ .

$4 a y^{2} \sqrt{3 a^{5}} \cdot (3 \sqrt{6 y^{4}})$

Step 2

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

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Step 2.1

Factor out $a^{4}$ .

$4 a y^{2} \sqrt{3 (a^{4} a)} \cdot (3 \sqrt{6 y^{4}})$

Step 2.2

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

$4 a y^{2} \sqrt{3 ({(a^{2})}^{2} a)} \cdot (3 \sqrt{6 y^{4}})$

Step 2.3

Reorder $3$ and ${(a^{2})}^{2}$ .

$4 a y^{2} \sqrt{{(a^{2})}^{2} \cdot 3 a} \cdot (3 \sqrt{6 y^{4}})$

Step 2.4

Add parentheses.

$4 a y^{2} \sqrt{{(a^{2})}^{2} \cdot (3 a)} \cdot (3 \sqrt{6 y^{4}})$

Step 3

Pull terms out from under the radical.

$4 a y^{2} (a^{2} \sqrt{3 a}) \cdot (3 \sqrt{6 y^{4}})$

Step 4

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

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Step 4.1

Move $a^{2}$ .

$4 (a^{2} a) y^{2} \sqrt{3 a} \cdot (3 \sqrt{6 y^{4}})$

Step 4.2

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

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Step 4.2.1

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

$4 (a^{2} a^{1}) y^{2} \sqrt{3 a} \cdot (3 \sqrt{6 y^{4}})$

Step 4.2.2

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

$4 a^{2 + 1} y^{2} \sqrt{3 a} \cdot (3 \sqrt{6 y^{4}})$

Step 4.3

Add $2$ and $1$ .

$4 a^{3} y^{2} \sqrt{3 a} \cdot (3 \sqrt{6 y^{4}})$

Step 5

Rewrite $6 y^{4}$ as ${(y^{2})}^{2} \cdot 6$ .

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Step 5.1

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

$4 a^{3} y^{2} \sqrt{3 a} \cdot (3 \sqrt{6 {(y^{2})}^{2}})$

Step 5.2

Reorder $6$ and ${(y^{2})}^{2}$ .

$4 a^{3} y^{2} \sqrt{3 a} \cdot (3 \sqrt{{(y^{2})}^{2} \cdot 6})$

Step 6

Pull terms out from under the radical.

$4 a^{3} y^{2} \sqrt{3 a} \cdot (3 (y^{2} \sqrt{6}))$

Step 7

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

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Step 7.1

Move $y^{2}$ .

$4 a^{3} (y^{2} y^{2}) \sqrt{3 a} \cdot (3 (\sqrt{6}))$

Step 7.2

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

$4 a^{3} y^{2 + 2} \sqrt{3 a} \cdot (3 (\sqrt{6}))$

Step 7.3

Add $2$ and $2$ .

$4 a^{3} y^{4} \sqrt{3 a} \cdot (3 (\sqrt{6}))$

Step 8

Multiply $4 a^{3} y^{4} \sqrt{3 a} (3 \sqrt{6})$ .

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Step 8.1

Multiply $3$ by $4$ .

$12 a^{3} y^{4} \sqrt{3 a} \sqrt{6}$

Step 8.2

Combine using the product rule for radicals.

$12 a^{3} y^{4} \sqrt{6 (3 a)}$

Step 8.3

Multiply $6$ by $3$ .

$12 a^{3} y^{4} \sqrt{18 a}$

Step 9

Rewrite $18 a$ as $3^{2} \cdot (2 a)$ .

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Step 9.1

Factor $9$ out of $18$ .

$12 a^{3} y^{4} \sqrt{9 (2) a}$

Step 9.2

Rewrite $9$ as $3^{2}$ .

$12 a^{3} y^{4} \sqrt{3^{2} \cdot 2 a}$

Step 9.3

Add parentheses.

$12 a^{3} y^{4} \sqrt{3^{2} \cdot (2 a)}$

Step 10

Pull terms out from under the radical.

$12 a^{3} y^{4} (3 \sqrt{2 a})$

Step 11

Multiply $3$ by $12$ .

$36 a^{3} y^{4} \sqrt{2 a}$