Finite Math Examples

$\frac{\sqrt{15}}{\sqrt[5]{20}}$

Step 1

Multiply $\frac{\sqrt{15}}{\sqrt[5]{20}}$ by $\frac{{\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{4}}$ .

$\frac{\sqrt{15}}{\sqrt[5]{20}} \cdot \frac{{\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{4}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{15}}{\sqrt[5]{20}}$ by $\frac{{\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{4}}$ .

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{\sqrt[5]{20} {\sqrt[5]{20}}^{4}}$

Step 2.2

Raise $\sqrt[5]{20}$ to the power of $1$ .

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{1} {\sqrt[5]{20}}^{4}}$

Step 2.3

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

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{1 + 4}}$

Step 2.4

Add $1$ and $4$ .

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{{\sqrt[5]{20}}^{5}}$

Step 2.5

Rewrite ${\sqrt[5]{20}}^{5}$ as $20$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{20}$ as $20^{\frac{1}{5}}$ .

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{{(20^{\frac{1}{5}})}^{5}}$

Step 2.5.2

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

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{20^{\frac{1}{5} \cdot 5}}$

Step 2.5.3

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

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{20^{\frac{5}{5}}}$

Step 2.5.4

Cancel the common factor of $5$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{20^{\frac{5}{5}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{20^{1}}$

Step 2.5.5

Evaluate the exponent.

$\frac{\sqrt{15} {\sqrt[5]{20}}^{4}}{20}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[5]{20}}^{4}$ as $\sqrt[5]{20^{4}}$ .

$\frac{\sqrt{15} \sqrt[5]{20^{4}}}{20}$

Step 3.2

Raise $20$ to the power of $4$ .

$\frac{\sqrt{15} \sqrt[5]{160000}}{20}$

Step 3.3

Rewrite $160000$ as $2^{5} \cdot 5000$ .

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Step 3.3.1

Factor $32$ out of $160000$ .

$\frac{\sqrt{15} \sqrt[5]{32 (5000)}}{20}$

Step 3.3.2

Rewrite $32$ as $2^{5}$ .

$\frac{\sqrt{15} \sqrt[5]{2^{5} \cdot 5000}}{20}$

Step 3.4

Pull terms out from under the radical.

$\frac{\sqrt{15} \cdot 2 \sqrt[5]{5000}}{20}$

Step 3.5

Combine exponents.

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Step 3.5.1

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

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Step 3.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{2 (15^{\frac{1}{2}} \sqrt[5]{5000})}{20}$

Step 3.5.1.2

Rewrite $15^{\frac{1}{2}}$ as $15^{\frac{5}{10}}$ .

$\frac{2 (15^{\frac{5}{10}} \sqrt[5]{5000})}{20}$

Step 3.5.1.3

Rewrite $15^{\frac{5}{10}}$ as $\sqrt[10]{15^{5}}$ .

$\frac{2 (\sqrt[10]{15^{5}} \sqrt[5]{5000})}{20}$

Step 3.5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{5000}$ as $5000^{\frac{1}{5}}$ .

$\frac{2 (\sqrt[10]{15^{5}} \cdot 5000^{\frac{1}{5}})}{20}$

Step 3.5.1.5

Rewrite $5000^{\frac{1}{5}}$ as $5000^{\frac{2}{10}}$ .

$\frac{2 (\sqrt[10]{15^{5}} \cdot 5000^{\frac{2}{10}})}{20}$

Step 3.5.1.6

Rewrite $5000^{\frac{2}{10}}$ as $\sqrt[10]{5000^{2}}$ .

$\frac{2 (\sqrt[10]{15^{5}} \sqrt[10]{5000^{2}})}{20}$

Step 3.5.2

Combine using the product rule for radicals.

$\frac{2 \sqrt[10]{15^{5} \cdot 5000^{2}}}{20}$

Step 3.5.3

Raise $15$ to the power of $5$ .

$\frac{2 \sqrt[10]{759375 \cdot 5000^{2}}}{20}$

Step 3.5.4

Raise $5000$ to the power of $2$ .

$\frac{2 \sqrt[10]{759375 \cdot 25000000}}{20}$

Step 3.5.5

Multiply $759375$ by $25000000$ .

$\frac{2 \sqrt[10]{18984375000000}}{20}$

Step 3.6

Rewrite $18984375000000$ as $5^{10} \cdot 1944000$ .

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Step 3.6.1

Factor $9765625$ out of $18984375000000$ .

$\frac{2 \sqrt[10]{9765625 (1944000)}}{20}$

Step 3.6.2

Rewrite $9765625$ as $5^{10}$ .

$\frac{2 \sqrt[10]{5^{10} \cdot 1944000}}{20}$

Step 3.7

Pull terms out from under the radical.

$\frac{2 (5 \sqrt[10]{1944000})}{20}$

Step 4

Multiply $2$ by $5$ .

$\frac{10 \sqrt[10]{1944000}}{20}$

Step 5

Cancel the common factor of $10$ and $20$ .

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Step 5.1

Factor $10$ out of $10 \sqrt[10]{1944000}$ .

$\frac{10 (\sqrt[10]{1944000})}{20}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $10$ out of $20$ .

$\frac{10 \sqrt[10]{1944000}}{10 \cdot 2}$

Step 5.2.2

Cancel the common factor.

$\frac{10 \sqrt[10]{1944000}}{10 \cdot 2}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[10]{1944000}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[10]{1944000}}{2}$

Decimal Form:

$2.12735334 \dots$