Precalculus Examples

$\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{- 4}}}$

Step 1

Combine $\sqrt{c^{2} d^{6}}$ and $\sqrt{4 c^{3} d^{- 4}}$ into a single radical.

$\sqrt{\frac{c^{2} d^{6}}{4 c^{3} d^{- 4}}}$

Step 2

Reduce the expression $\frac{c^{2} d^{6}}{4 c^{3} d^{- 4}}$ by cancelling the common factors.

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Step 2.1

Factor $c^{2}$ out of $c^{2} d^{6}$ .

$\sqrt{\frac{c^{2} (d^{6})}{4 c^{3} d^{- 4}}}$

Step 2.2

Factor $c^{2}$ out of $4 c^{3} d^{- 4}$ .

$\sqrt{\frac{c^{2} (d^{6})}{c^{2} (4 c d^{- 4})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{c^{2} d^{6}}{c^{2} (4 c d^{- 4})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{d^{6}}{4 c d^{- 4}}}$

Step 3

Move $d^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt{\frac{d^{6} d^{4}}{4 c}}$

Step 4

Multiply $d^{6}$ by $d^{4}$ by adding the exponents.

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Step 4.1

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

$\sqrt{\frac{d^{6 + 4}}{4 c}}$

Step 4.2

Add $6$ and $4$ .

$\sqrt{\frac{d^{10}}{4 c}}$

Step 5

Rewrite $\frac{d^{10}}{4 c}$ as ${(\frac{d^{5}}{2})}^{2} \frac{1}{c}$ .

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Step 5.1

Factor the perfect power ${(d^{5})}^{2}$ out of $d^{10}$ .

$\sqrt{\frac{{(d^{5})}^{2} \cdot 1}{4 c}}$

Step 5.2

Factor the perfect power $2^{2}$ out of $4 c$ .

$\sqrt{\frac{{(d^{5})}^{2} \cdot 1}{2^{2} c}}$

Step 5.3

Rearrange the fraction $\frac{{(d^{5})}^{2} \cdot 1}{2^{2} c}$ .

$\sqrt{{(\frac{d^{5}}{2})}^{2} \frac{1}{c}}$

Step 6

Pull terms out from under the radical.

$\frac{d^{5}}{2} \sqrt{\frac{1}{c}}$

Step 7

Rewrite $\sqrt{\frac{1}{c}}$ as $\frac{\sqrt{1}}{\sqrt{c}}$ .

$\frac{d^{5}}{2} \cdot \frac{\sqrt{1}}{\sqrt{c}}$

Step 8

Combine.

$\frac{d^{5} \sqrt{1}}{2 \sqrt{c}}$

Step 9

Any root of $1$ is $1$ .

$\frac{d^{5} \cdot 1}{2 \sqrt{c}}$

Step 10

Multiply $d^{5}$ by $1$ .

$\frac{d^{5}}{2 \sqrt{c}}$

Step 11

Multiply $\frac{d^{5}}{2 \sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{d^{5}}{2 \sqrt{c}} \cdot \frac{\sqrt{c}}{\sqrt{c}}$

Step 12

Combine and simplify the denominator.

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Step 12.1

Multiply $\frac{d^{5}}{2 \sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{d^{5} \sqrt{c}}{2 \sqrt{c} \sqrt{c}}$

Step 12.2

Move $\sqrt{c}$ .

$\frac{d^{5} \sqrt{c}}{2 (\sqrt{c} \sqrt{c})}$

Step 12.3

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

$\frac{d^{5} \sqrt{c}}{2 ({\sqrt{c}}^{1} \sqrt{c})}$

Step 12.4

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

$\frac{d^{5} \sqrt{c}}{2 ({\sqrt{c}}^{1} {\sqrt{c}}^{1})}$

Step 12.5

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

$\frac{d^{5} \sqrt{c}}{2 {\sqrt{c}}^{1 + 1}}$

Step 12.6

Add $1$ and $1$ .

$\frac{d^{5} \sqrt{c}}{2 {\sqrt{c}}^{2}}$

Step 12.7

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

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Step 12.7.1

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

$\frac{d^{5} \sqrt{c}}{2 {(c^{\frac{1}{2}})}^{2}}$

Step 12.7.2

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

$\frac{d^{5} \sqrt{c}}{2 c^{\frac{1}{2} \cdot 2}}$

Step 12.7.3

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

$\frac{d^{5} \sqrt{c}}{2 c^{\frac{2}{2}}}$

Step 12.7.4

Cancel the common factor of $2$ .

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Step 12.7.4.1

Cancel the common factor.

$\frac{d^{5} \sqrt{c}}{2 c^{\frac{2}{2}}}$

Step 12.7.4.2

Rewrite the expression.

$\frac{d^{5} \sqrt{c}}{2 c^{1}}$

Step 12.7.5

Simplify.

$\frac{d^{5} \sqrt{c}}{2 c}$