Finite Math Examples

$\frac{\sqrt{3}}{\sqrt[3]{5}}$

Step 1

Multiply $\frac{\sqrt{3}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{\sqrt{3}}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{3}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 2.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

Step 2.3

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

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Step 2.4

Add $1$ and $2$ .

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Step 2.5

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

Step 2.5.2

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

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

Step 2.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{5^{1}}$

Step 2.5.5

Evaluate the exponent.

$\frac{\sqrt{3} {\sqrt[3]{5}}^{2}}{5}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$\frac{\sqrt{3} \sqrt[3]{5^{2}}}{5}$

Step 3.2

Raise $5$ to the power of $2$ .

$\frac{\sqrt{3} \sqrt[3]{25}}{5}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $6$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3^{\frac{1}{2}} \sqrt[3]{25}}{5}$

Step 4.1.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{3}{6}}$ .

$\frac{3^{\frac{3}{6}} \sqrt[3]{25}}{5}$

Step 4.1.3

Rewrite $3^{\frac{3}{6}}$ as $\sqrt[6]{3^{3}}$ .

$\frac{\sqrt[6]{3^{3}} \sqrt[3]{25}}{5}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$\frac{\sqrt[6]{3^{3}} \cdot 25^{\frac{1}{3}}}{5}$

Step 4.1.5

Rewrite $25^{\frac{1}{3}}$ as $25^{\frac{2}{6}}$ .

$\frac{\sqrt[6]{3^{3}} \cdot 25^{\frac{2}{6}}}{5}$

Step 4.1.6

Rewrite $25^{\frac{2}{6}}$ as $\sqrt[6]{25^{2}}$ .

$\frac{\sqrt[6]{3^{3}} \sqrt[6]{25^{2}}}{5}$

Step 4.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{3^{3} \cdot 25^{2}}}{5}$

Step 4.3

Raise $3$ to the power of $3$ .

$\frac{\sqrt[6]{27 \cdot 25^{2}}}{5}$

Step 4.4

Raise $25$ to the power of $2$ .

$\frac{\sqrt[6]{27 \cdot 625}}{5}$

Step 5

Multiply $27$ by $625$ .

$\frac{\sqrt[6]{16875}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[6]{16875}}{5}$

Decimal Form:

$1.01290945 \dots$