Algebra Examples

$3 \sqrt{8 a^{3}} \div \sqrt{b^{3}}$

Step 1

Rewrite the division as a fraction.

$\frac{3 \sqrt{8 a^{3}}}{\sqrt{b^{3}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $8 a^{3}$ as ${(2 a)}^{2} \cdot (2 a)$ .

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Step 2.1.1

Factor $4$ out of $8$ .

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

Step 2.1.2

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

$\frac{3 \sqrt{2^{2} \cdot 2 a^{3}}}{\sqrt{b^{3}}}$

Step 2.1.3

Factor out $a^{2}$ .

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

Step 2.1.4

Move $2$ .

$\frac{3 \sqrt{2^{2} a^{2} \cdot 2 a}}{\sqrt{b^{3}}}$

Step 2.1.5

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

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

Step 2.1.6

Add parentheses.

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

Step 2.2

Pull terms out from under the radical.

$\frac{3 \cdot 2 a \sqrt{2 a}}{\sqrt{b^{3}}}$

Step 2.3

Multiply $3$ by $2$ .

$\frac{6 a \sqrt{2 a}}{\sqrt{b^{3}}}$

Step 3

Simplify the denominator.

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Step 3.1

Factor out $b^{2}$ .

$\frac{6 a \sqrt{2 a}}{\sqrt{b^{2} b}}$

Step 3.2

Pull terms out from under the radical.

$\frac{6 a \sqrt{2 a}}{b \sqrt{b}}$

Step 4

Multiply $\frac{6 a \sqrt{2 a}}{b \sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{6 a \sqrt{2 a}}{b \sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}}$

Step 5

Combine fractions.

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Step 5.1

Combine and simplify the denominator.

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Step 5.1.1

Multiply $\frac{6 a \sqrt{2 a}}{b \sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b \sqrt{b} \sqrt{b}}$

Step 5.1.2

Move $\sqrt{b}$ .

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b (\sqrt{b} \sqrt{b})}$

Step 5.1.3

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

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b ({\sqrt{b}}^{1} \sqrt{b})}$

Step 5.1.4

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

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b ({\sqrt{b}}^{1} {\sqrt{b}}^{1})}$

Step 5.1.5

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

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b {\sqrt{b}}^{1 + 1}}$

Step 5.1.6

Add $1$ and $1$ .

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b {\sqrt{b}}^{2}}$

Step 5.1.7

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

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Step 5.1.7.1

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

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b {(b^{\frac{1}{2}})}^{2}}$

Step 5.1.7.2

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

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b \cdot b^{\frac{1}{2} \cdot 2}}$

Step 5.1.7.3

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

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b \cdot b^{\frac{2}{2}}}$

Step 5.1.7.4

Cancel the common factor of $2$ .

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Step 5.1.7.4.1

Cancel the common factor.

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b \cdot b^{\frac{2}{2}}}$

Step 5.1.7.4.2

Rewrite the expression.

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b \cdot b^{1}}$

Step 5.1.7.5

Simplify.

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b \cdot b}$

Step 5.2

Multiply $b$ by $b$ .

$\frac{6 a \sqrt{2 a} \sqrt{b}}{b^{2}}$

Step 6

Combine using the product rule for radicals.

$\frac{6 a \sqrt{b \cdot 2 a}}{b^{2}}$

Step 7

Move $2$ to the left of $b$ .

$\frac{6 a \sqrt{2 b a}}{b^{2}}$