Finite Math Examples

$\frac{\sqrt{c d^{6}}}{\sqrt[5]{c d}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $c d^{6}$ as ${(d^{3})}^{2} c$ .

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Step 1.1.1

Rewrite $d^{6}$ as ${(d^{3})}^{2}$ .

$\frac{\sqrt{c {(d^{3})}^{2}}}{\sqrt[5]{c d}}$

Step 1.1.2

Reorder $c$ and ${(d^{3})}^{2}$ .

$\frac{\sqrt{{(d^{3})}^{2} c}}{\sqrt[5]{c d}}$

Step 1.2

Pull terms out from under the radical.

$\frac{d^{3} \sqrt{c}}{\sqrt[5]{c d}}$

Step 2

Multiply $\frac{d^{3} \sqrt{c}}{\sqrt[5]{c d}}$ by $\frac{{\sqrt[5]{c d}}^{4}}{{\sqrt[5]{c d}}^{4}}$ .

$\frac{d^{3} \sqrt{c}}{\sqrt[5]{c d}} \cdot \frac{{\sqrt[5]{c d}}^{4}}{{\sqrt[5]{c d}}^{4}}$

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

Multiply $\frac{d^{3} \sqrt{c}}{\sqrt[5]{c d}}$ by $\frac{{\sqrt[5]{c d}}^{4}}{{\sqrt[5]{c d}}^{4}}$ .

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{\sqrt[5]{c d} {\sqrt[5]{c d}}^{4}}$

Step 3.1.2

Raise $\sqrt[5]{c d}$ to the power of $1$ .

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{\sqrt[5]{c d}}^{1} {\sqrt[5]{c d}}^{4}}$

Step 3.1.3

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

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{\sqrt[5]{c d}}^{1 + 4}}$

Step 3.1.4

Add $1$ and $4$ .

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{\sqrt[5]{c d}}^{5}}$

Step 3.1.5

Rewrite ${\sqrt[5]{c d}}^{5}$ as $c d$ .

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Step 3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{c d}$ as ${(c d)}^{\frac{1}{5}}$ .

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{({(c d)}^{\frac{1}{5}})}^{5}}$

Step 3.1.5.2

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

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{(c d)}^{\frac{1}{5} \cdot 5}}$

Step 3.1.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{(c d)}^{\frac{5}{5}}}$

Step 3.1.5.4

Cancel the common factor of $5$ .

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Step 3.1.5.4.1

Cancel the common factor.

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{(c d)}^{\frac{5}{5}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{{(c d)}^{1}}$

Step 3.1.5.5

Simplify.

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{c d}$

Step 3.2

Cancel the common factor of $d^{3}$ and $d$ .

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Step 3.2.1

Factor $d$ out of $d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}$ .

$\frac{d (d^{2} \sqrt{c} {\sqrt[5]{c d}}^{4})}{c d}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $d$ out of $c d$ .

$\frac{d (d^{2} \sqrt{c} {\sqrt[5]{c d}}^{4})}{d c}$

Step 3.2.2.2

Cancel the common factor.

$\frac{d (d^{2} \sqrt{c} {\sqrt[5]{c d}}^{4})}{d c}$

Step 3.2.2.3

Rewrite the expression.

$\frac{d^{2} \sqrt{c} {\sqrt[5]{c d}}^{4}}{c}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[5]{c d}}^{4}$ as $\sqrt[5]{{(c d)}^{4}}$ .

$\frac{d^{2} \sqrt{c} \sqrt[5]{{(c d)}^{4}}}{c}$

Step 4.2

Apply the product rule to $c d$ .

$\frac{d^{2} \sqrt{c} \sqrt[5]{c^{4} d^{4}}}{c}$

Step 4.3

Combine exponents.

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Step 4.3.1

Rewrite the expression using the least common index of $10$ .

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Step 4.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{c^{4} d^{4}}$ as ${(c^{4} d^{4})}^{\frac{1}{5}}$ .

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

Step 4.3.1.2

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

$\frac{d^{2} ({(c^{4} d^{4})}^{\frac{2}{10}} \sqrt{c})}{c}$

Step 4.3.1.3

Rewrite ${(c^{4} d^{4})}^{\frac{2}{10}}$ as $\sqrt[10]{{(c^{4} d^{4})}^{2}}$ .

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} \sqrt{c})}{c}$

Step 4.3.1.4

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

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} c^{\frac{1}{2}})}{c}$

Step 4.3.1.5

Rewrite $c^{\frac{1}{2}}$ as $c^{\frac{5}{10}}$ .

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} c^{\frac{5}{10}})}{c}$

Step 4.3.1.6

Rewrite $c^{\frac{5}{10}}$ as $\sqrt[10]{c^{5}}$ .

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} \sqrt[10]{c^{5}})}{c}$

Step 4.3.2

Combine using the product rule for radicals.

$\frac{d^{2} \sqrt[10]{{(c^{4} d^{4})}^{2} c^{5}}}{c}$

Step 4.3.3

Apply the product rule to $c^{4} d^{4}$ .

$\frac{d^{2} \sqrt[10]{{(c^{4})}^{2} {(d^{4})}^{2} c^{5}}}{c}$

Step 4.3.4

Multiply the exponents in ${(c^{4})}^{2}$ .

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Step 4.3.4.1

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

$\frac{d^{2} \sqrt[10]{c^{4 \cdot 2} {(d^{4})}^{2} c^{5}}}{c}$

Step 4.3.4.2

Multiply $4$ by $2$ .

$\frac{d^{2} \sqrt[10]{c^{8} {(d^{4})}^{2} c^{5}}}{c}$

Step 4.3.5

Multiply the exponents in ${(d^{4})}^{2}$ .

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Step 4.3.5.1

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

$\frac{d^{2} \sqrt[10]{c^{8} d^{4 \cdot 2} c^{5}}}{c}$

Step 4.3.5.2

Multiply $4$ by $2$ .

$\frac{d^{2} \sqrt[10]{c^{8} d^{8} c^{5}}}{c}$

Step 4.3.6

Multiply $c^{8}$ by $c^{5}$ by adding the exponents.

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Step 4.3.6.1

Move $c^{5}$ .

$\frac{d^{2} \sqrt[10]{c^{5} c^{8} d^{8}}}{c}$

Step 4.3.6.2

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

$\frac{d^{2} \sqrt[10]{c^{5 + 8} d^{8}}}{c}$

Step 4.3.6.3

Add $5$ and $8$ .

$\frac{d^{2} \sqrt[10]{c^{13} d^{8}}}{c}$

Step 4.4

Rewrite $c^{13} d^{8}$ as $c^{10} (c^{3} d^{8})$ .

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Step 4.4.1

Factor out $c^{10}$ .

$\frac{d^{2} \sqrt[10]{c^{10} c^{3} d^{8}}}{c}$

Step 4.4.2

Add parentheses.

$\frac{d^{2} \sqrt[10]{c^{10} (c^{3} d^{8})}}{c}$

Step 4.5

Pull terms out from under the radical.

$\frac{d^{2} (c \sqrt[10]{c^{3} d^{8}})}{c}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite.

$\frac{d^{3} \sqrt{c} {\sqrt[5]{c d}}^{4}}{c}$

Step 5.2

Factor $d$ out of $d^{3}$ .

$\frac{d \cdot d^{2} \sqrt{c} {\sqrt[5]{c d}}^{4}}{c}$

Step 5.3

Rewrite.

$\frac{d^{2} \sqrt{c} {\sqrt[5]{c d}}^{4}}{c}$

Step 5.4

Rewrite ${\sqrt[5]{c d}}^{4}$ as $\sqrt[5]{{(c d)}^{4}}$ .

$\frac{d^{2} \sqrt{c} \sqrt[5]{{(c d)}^{4}}}{c}$

Step 5.5

Apply the product rule to $c d$ .

$\frac{d^{2} \sqrt{c} \sqrt[5]{c^{4} d^{4}}}{c}$

Step 5.6

Combine exponents.

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Step 5.6.1

Rewrite the expression using the least common index of $10$ .

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Step 5.6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{c^{4} d^{4}}$ as ${(c^{4} d^{4})}^{\frac{1}{5}}$ .

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

Step 5.6.1.2

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

$\frac{d^{2} ({(c^{4} d^{4})}^{\frac{2}{10}} \sqrt{c})}{c}$

Step 5.6.1.3

Rewrite ${(c^{4} d^{4})}^{\frac{2}{10}}$ as $\sqrt[10]{{(c^{4} d^{4})}^{2}}$ .

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} \sqrt{c})}{c}$

Step 5.6.1.4

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

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} c^{\frac{1}{2}})}{c}$

Step 5.6.1.5

Rewrite $c^{\frac{1}{2}}$ as $c^{\frac{5}{10}}$ .

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} c^{\frac{5}{10}})}{c}$

Step 5.6.1.6

Rewrite $c^{\frac{5}{10}}$ as $\sqrt[10]{c^{5}}$ .

$\frac{d^{2} (\sqrt[10]{{(c^{4} d^{4})}^{2}} \sqrt[10]{c^{5}})}{c}$

Step 5.6.2

Combine using the product rule for radicals.

$\frac{d^{2} \sqrt[10]{{(c^{4} d^{4})}^{2} c^{5}}}{c}$

Step 5.6.3

Apply the product rule to $c^{4} d^{4}$ .

$\frac{d^{2} \sqrt[10]{{(c^{4})}^{2} {(d^{4})}^{2} c^{5}}}{c}$

Step 5.6.4

Multiply the exponents in ${(c^{4})}^{2}$ .

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Step 5.6.4.1

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

$\frac{d^{2} \sqrt[10]{c^{4 \cdot 2} {(d^{4})}^{2} c^{5}}}{c}$

Step 5.6.4.2

Multiply $4$ by $2$ .

$\frac{d^{2} \sqrt[10]{c^{8} {(d^{4})}^{2} c^{5}}}{c}$

Step 5.6.5

Multiply the exponents in ${(d^{4})}^{2}$ .

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Step 5.6.5.1

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

$\frac{d^{2} \sqrt[10]{c^{8} d^{4 \cdot 2} c^{5}}}{c}$

Step 5.6.5.2

Multiply $4$ by $2$ .

$\frac{d^{2} \sqrt[10]{c^{8} d^{8} c^{5}}}{c}$

Step 5.6.6

Multiply $c^{8}$ by $c^{5}$ by adding the exponents.

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Step 5.6.6.1

Move $c^{5}$ .

$\frac{d^{2} \sqrt[10]{c^{5} c^{8} d^{8}}}{c}$

Step 5.6.6.2

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

$\frac{d^{2} \sqrt[10]{c^{5 + 8} d^{8}}}{c}$

Step 5.6.6.3

Add $5$ and $8$ .

$\frac{d^{2} \sqrt[10]{c^{13} d^{8}}}{c}$

Step 5.7

Rewrite $c^{13} d^{8}$ as $c^{10} (c^{3} d^{8})$ .

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Step 5.7.1

Factor out $c^{10}$ .

$\frac{d^{2} \sqrt[10]{c^{10} c^{3} d^{8}}}{c}$

Step 5.7.2

Add parentheses.

$\frac{d^{2} \sqrt[10]{c^{10} (c^{3} d^{8})}}{c}$

Step 5.8

Pull terms out from under the radical.

$\frac{d^{2} (c \sqrt[10]{c^{3} d^{8}})}{c}$

Step 5.9

Remove unnecessary parentheses.

$\frac{d^{2} c \sqrt[10]{c^{3} d^{8}}}{c}$

Step 6

Cancel the common factor of $c$ .

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Step 6.1

Cancel the common factor.

$\frac{d^{2} c \sqrt[10]{c^{3} d^{8}}}{c}$

Step 6.2

Divide $d^{2} \sqrt[10]{c^{3} d^{8}}$ by $1$ .

$d^{2} \sqrt[10]{c^{3} d^{8}}$