Precalculus Examples

$\sqrt{5 a^{3} b} \cdot \sqrt{10 a b}$

Step 1

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

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Step 1.1

Factor out $a^{2}$ .

$\sqrt{5 (a^{2} a) b} \cdot \sqrt{10 a b}$

Step 1.2

Reorder $5$ and $a^{2}$ .

$\sqrt{a^{2} \cdot 5 a b} \cdot \sqrt{10 a b}$

Step 1.3

Add parentheses.

$\sqrt{a^{2} \cdot 5 (a b)} \cdot \sqrt{10 a b}$

Step 1.4

Add parentheses.

$\sqrt{a^{2} \cdot (5 a b)} \cdot \sqrt{10 a b}$

Step 2

Pull terms out from under the radical.

$a \sqrt{5 a b} \cdot \sqrt{10 a b}$

Step 3

Multiply $a \sqrt{5 a b} \sqrt{10 a b}$ .

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Step 3.1

Combine using the product rule for radicals.

$a \sqrt{10 a b (5 a b)}$

Step 3.2

Multiply $5$ by $10$ .

$a \sqrt{50 a b (a b)}$

Step 3.3

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

$a \sqrt{50 (a^{1} a) b \cdot b}$

Step 3.4

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

$a \sqrt{50 (a^{1} a^{1}) b \cdot b}$

Step 3.5

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

$a \sqrt{50 a^{1 + 1} b \cdot b}$

Step 3.6

Add $1$ and $1$ .

$a \sqrt{50 a^{2} b \cdot b}$

Step 3.7

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

$a \sqrt{50 a^{2} (b^{1} b)}$

Step 3.8

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

$a \sqrt{50 a^{2} (b^{1} b^{1})}$

Step 3.9

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

$a \sqrt{50 a^{2} b^{1 + 1}}$

Step 3.10

Add $1$ and $1$ .

$a \sqrt{50 a^{2} b^{2}}$

Step 4

Rewrite $50 a^{2} b^{2}$ as ${(5 a b)}^{2} \cdot 2$ .

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Step 4.1

Factor $25$ out of $50$ .

$a \sqrt{25 (2) a^{2} b^{2}}$

Step 4.2

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

$a \sqrt{5^{2} \cdot 2 a^{2} b^{2}}$

Step 4.3

Move $2$ .

$a \sqrt{5^{2} a^{2} b^{2} \cdot 2}$

Step 4.4

Rewrite $5^{2} a^{2} b^{2}$ as ${(5 a b)}^{2}$ .

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

Step 5

Pull terms out from under the radical.

$a (5 a b \sqrt{2})$

Step 6

Rewrite using the commutative property of multiplication.

$5 \sqrt{2} a (a b)$

Step 7

Multiply $a$ by $a$ by adding the exponents.

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Step 7.1

Move $a$ .

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

Step 7.2

Multiply $a$ by $a$ .

$5 \sqrt{2} a^{2} b$