Precalculus Examples

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

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

Step 1

Rewrite

5 a^{3} b

$5 a^{3} b$ as

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

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

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Step 1.1

Factor out

a^{2}

$a^{2}$ .

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

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

Step 1.2

Reorder

5

$5$ and

a^{2}

$a^{2}$ .

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

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

Step 1.3

Add parentheses.

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

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

Step 1.4

Add parentheses.

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

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

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

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

Step 2

Pull terms out from under the radical.

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

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

Step 3

Multiply

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

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

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Step 3.1

Combine using the product rule for radicals.

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

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

Step 3.2

Multiply

5

$5$ by

10

$10$ .

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

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

Step 3.3

Raise

a

$a$ to the power of

1

$1$ .

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

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

Step 3.4

Raise

a

$a$ to the power of

1

$1$ .

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

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

Step 3.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.6

Add

1

$1$ and

1

$1$ .

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

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

Step 3.7

Raise

b

$b$ to the power of

1

$1$ .

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

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

Step 3.8

Raise

b

$b$ to the power of

1

$1$ .

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

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

Step 3.9

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.10

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

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

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

$a$ .

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

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

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$