Basic Math Examples

$5 \sqrt{a^{9} b} \cdot (5 \sqrt{a^{6} b})$

Step 1

Rewrite $a^{9} b$ as ${({(a^{2})}^{2})}^{2} (a b)$ .

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Step 1.1

Factor out $a^{8}$ .

$5 \sqrt{a^{8} a b} \cdot (5 \sqrt{a^{6} b})$

Step 1.2

Rewrite $a^{8}$ as ${(a^{4})}^{2}$ .

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

Step 1.3

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

$5 \sqrt{{({(a^{2})}^{2})}^{2} a b} \cdot (5 \sqrt{a^{6} b})$

Step 1.4

Add parentheses.

$5 \sqrt{{({(a^{2})}^{2})}^{2} (a b)} \cdot (5 \sqrt{a^{6} b})$

Step 2

Pull terms out from under the radical.

$5 ({(a^{2})}^{2} \sqrt{a b}) \cdot (5 \sqrt{a^{6} b})$

Step 3

Apply basic rules of exponents.

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Step 3.1

Multiply the exponents in ${(a^{2})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 (a^{2 \cdot 2} \sqrt{a b}) \cdot (5 \sqrt{a^{6} b})$

Step 3.1.2

Multiply $2$ by $2$ .

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

Step 3.2

Rewrite $a^{6}$ as ${(a^{3})}^{2}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $a^{4}$ by $a^{3}$ by adding the exponents.

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Step 5.1

Move $a^{3}$ .

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

Step 5.2

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

$5 a^{3 + 4} \sqrt{a b} \cdot (5 (\sqrt{b}))$

Step 5.3

Add $3$ and $4$ .

$5 a^{7} \sqrt{a b} \cdot (5 (\sqrt{b}))$

Step 6

Multiply $5 a^{7} \sqrt{a b} (5 \sqrt{b})$ .

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Step 6.1

Multiply $5$ by $5$ .

$25 a^{7} \sqrt{a b} \sqrt{b}$

Step 6.2

Combine using the product rule for radicals.

$25 a^{7} \sqrt{b (a b)}$

Step 6.3

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

$25 a^{7} \sqrt{a (b^{1} b)}$

Step 6.4

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

$25 a^{7} \sqrt{a (b^{1} b^{1})}$

Step 6.5

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

$25 a^{7} \sqrt{a b^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$25 a^{7} \sqrt{a b^{2}}$

Step 7

Reorder $a$ and $b^{2}$ .

$25 a^{7} \sqrt{b^{2} a}$

Step 8

Pull terms out from under the radical.

$25 a^{7} (b \sqrt{a})$

Step 9

Reorder factors in $25 a^{7} (b \sqrt{a})$ .

$25 a^{7} b \sqrt{a}$