Precalculus Examples

$\sqrt{8 a^{5}} (- \sqrt{2 a^{3}})$

Step 1

Rewrite using the commutative property of multiplication.

$- \sqrt{8 a^{5}} \sqrt{2 a^{3}}$

Step 2

Rewrite $8 a^{5}$ as ${(2 a^{2})}^{2} \cdot (2 a)$ .

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Step 2.1

Factor $4$ out of $8$ .

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

Step 2.2

Rewrite $4$ as $2^{2}$ .

$- \sqrt{2^{2} \cdot 2 a^{5}} \sqrt{2 a^{3}}$

Step 2.3

Factor out $a^{4}$ .

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

Step 2.4

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

$- \sqrt{2^{2} \cdot 2 ({(a^{2})}^{2} a)} \sqrt{2 a^{3}}$

Step 2.5

Move $2$ .

$- \sqrt{2^{2} {(a^{2})}^{2} \cdot 2 a} \sqrt{2 a^{3}}$

Step 2.6

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

$- \sqrt{{(2 a^{2})}^{2} \cdot 2 a} \sqrt{2 a^{3}}$

Step 2.7

Add parentheses.

$- \sqrt{{(2 a^{2})}^{2} \cdot (2 a)} \sqrt{2 a^{3}}$

Step 3

Pull terms out from under the radical.

$- (2 a^{2} \sqrt{2 a}) \sqrt{2 a^{3}}$

Step 4

Multiply $2$ by $- 1$ .

$- 2 (a^{2} \sqrt{2 a}) \sqrt{2 a^{3}}$

Step 5

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

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Step 5.1

Factor out $a^{2}$ .

$- 2 a^{2} \sqrt{2 a} \sqrt{2 (a^{2} a)}$

Step 5.2

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

$- 2 a^{2} \sqrt{2 a} \sqrt{a^{2} \cdot 2 a}$

Step 5.3

Add parentheses.

$- 2 a^{2} \sqrt{2 a} \sqrt{a^{2} \cdot (2 a)}$

Step 6

Pull terms out from under the radical.

$- 2 a^{2} \sqrt{2 a} (a \sqrt{2 a})$

Step 7

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

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Step 7.1

Move $a$ .

$- 2 (a \cdot a^{2}) \sqrt{2 a} \sqrt{2 a}$

Step 7.2

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

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Step 7.2.1

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

$- 2 (a^{1} a^{2}) \sqrt{2 a} \sqrt{2 a}$

Step 7.2.2

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

$- 2 a^{1 + 2} \sqrt{2 a} \sqrt{2 a}$

Step 7.3

Add $1$ and $2$ .

$- 2 a^{3} \sqrt{2 a} \sqrt{2 a}$

Step 8

Multiply $- 2 a^{3} \sqrt{2 a} \sqrt{2 a}$ .

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Step 8.1

Raise $\sqrt{2 a}$ to the power of $1$ .

$- 2 a^{3} ({\sqrt{2 a}}^{1} \sqrt{2 a})$

Step 8.2

Raise $\sqrt{2 a}$ to the power of $1$ .

$- 2 a^{3} ({\sqrt{2 a}}^{1} {\sqrt{2 a}}^{1})$

Step 8.3

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

$- 2 a^{3} {\sqrt{2 a}}^{1 + 1}$

Step 8.4

Add $1$ and $1$ .

$- 2 a^{3} {\sqrt{2 a}}^{2}$

Step 9

Rewrite ${\sqrt{2 a}}^{2}$ as $2 a$ .

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Step 9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 a}$ as ${(2 a)}^{\frac{1}{2}}$ .

$- 2 a^{3} {({(2 a)}^{\frac{1}{2}})}^{2}$

Step 9.2

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

$- 2 a^{3} {(2 a)}^{\frac{1}{2} \cdot 2}$

Step 9.3

Combine $\frac{1}{2}$ and $2$ .

$- 2 a^{3} {(2 a)}^{\frac{2}{2}}$

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

$- 2 a^{3} {(2 a)}^{\frac{2}{2}}$

Step 9.4.2

Rewrite the expression.

$- 2 a^{3} {(2 a)}^{1}$

Step 9.5

Simplify.

$- 2 a^{3} (2 a)$

Step 10

Rewrite using the commutative property of multiplication.

$- 2 \cdot 2 a^{3} a$

Step 11

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

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Step 11.1

Move $a$ .

$- 2 \cdot 2 (a \cdot a^{3})$

Step 11.2

Multiply $a$ by $a^{3}$ .

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Step 11.2.1

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

$- 2 \cdot 2 (a^{1} a^{3})$

Step 11.2.2

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

$- 2 \cdot 2 a^{1 + 3}$

Step 11.3

Add $1$ and $3$ .

$- 2 \cdot 2 a^{4}$

Step 12

Multiply $- 2$ by $2$ .

$- 4 a^{4}$