Basic Math Examples

$\sqrt{\frac{a b c^{- 2}}{8 b^{2} c}}$

Step 1

Reduce the expression $\frac{a b c^{- 2}}{8 b^{2} c}$ by cancelling the common factors.

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Step 1.1

Factor $b$ out of $a b c^{- 2}$ .

$\sqrt{\frac{b (a c^{- 2})}{8 b^{2} c}}$

Step 1.2

Factor $b$ out of $8 b^{2} c$ .

$\sqrt{\frac{b (a c^{- 2})}{b (8 b c)}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{b (a c^{- 2})}{b (8 b c)}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{a c^{- 2}}{8 b c}}$

Step 2

Move $c^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt{\frac{a}{8 b c \cdot c^{2}}}$

Step 3

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

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Step 3.1

Move $c^{2}$ .

$\sqrt{\frac{a}{8 b (c^{2} c)}}$

Step 3.2

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

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Step 3.2.1

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

$\sqrt{\frac{a}{8 b (c^{2} c^{1})}}$

Step 3.2.2

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

$\sqrt{\frac{a}{8 b c^{2 + 1}}}$

Step 3.3

Add $2$ and $1$ .

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

Step 4

Rewrite $\frac{a}{8 b c^{3}}$ as ${(\frac{1}{2 c})}^{2} \frac{a}{2 b c}$ .

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Step 4.1

Factor the perfect power $1^{2}$ out of $a$ .

$\sqrt{\frac{1^{2} a}{8 b c^{3}}}$

Step 4.2

Rearrange the fraction $\frac{1^{2} a}{{(2 c)}^{2} \cdot (2 b c)}$ .

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

Step 5

Pull terms out from under the radical.

$\frac{1}{2 c} \sqrt{\frac{a}{2 b c}}$

Step 6

Rewrite $\sqrt{\frac{a}{2 b c}}$ as $\frac{\sqrt{a}}{\sqrt{2 b c}}$ .

$\frac{1}{2 c} \cdot \frac{\sqrt{a}}{\sqrt{2 b c}}$

Step 7

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

$\frac{1}{2 c} (\frac{\sqrt{a}}{\sqrt{2 b c}} \cdot \frac{\sqrt{2 b c}}{\sqrt{2 b c}})$

Step 8

Combine and simplify the denominator.

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Step 8.1

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

$\frac{1}{2 c} \cdot \frac{\sqrt{a} \sqrt{2 b c}}{\sqrt{2 b c} \sqrt{2 b c}}$

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Simplify.

$\frac{1}{2 c} \cdot \frac{\sqrt{a} \sqrt{2 b c}}{2 b c}$

Step 9

Combine using the product rule for radicals.

$\frac{1}{2 c} \cdot \frac{\sqrt{a \cdot 2 b c}}{2 b c}$

Step 10

Multiply $\frac{1}{2 c} \cdot \frac{\sqrt{a \cdot 2 b c}}{2 b c}$ .

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Step 10.1

Multiply $\frac{1}{2 c}$ by $\frac{\sqrt{a \cdot 2 b c}}{2 b c}$ .

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

Step 10.2

Multiply $2$ by $2$ .

$\frac{\sqrt{a \cdot 2 b c}}{4 c (b c)}$

Step 10.3

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

$\frac{\sqrt{a \cdot 2 b c}}{4 (b (c^{1} c))}$

Step 10.4

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

$\frac{\sqrt{a \cdot 2 b c}}{4 (b (c^{1} c^{1}))}$

Step 10.5

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

$\frac{\sqrt{a \cdot 2 b c}}{4 (b c^{1 + 1})}$

Step 10.6

Add $1$ and $1$ .

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

Step 11

Reorder factors in $\frac{\sqrt{a \cdot 2 b c}}{4 b c^{2}}$ .

$\frac{\sqrt{2 a b c}}{4 b c^{2}}$