Finite Math Examples

$\sqrt{\frac{4}{9}} \div \sqrt{\frac{3}{5}}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt{\frac{4}{9}}}{\sqrt{\frac{3}{5}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$\frac{\frac{\sqrt{4}}{\sqrt{9}}}{\sqrt{\frac{3}{5}}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{\sqrt{2^{2}}}{\sqrt{9}}}{\sqrt{\frac{3}{5}}}$

Step 2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\frac{2}{\sqrt{9}}}{\sqrt{\frac{3}{5}}}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\frac{2}{3}}{\sqrt{\frac{3}{5}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $\sqrt{\frac{3}{5}}$ as $\frac{\sqrt{3}}{\sqrt{5}}$ .

$\frac{\frac{2}{3}}{\frac{\sqrt{3}}{\sqrt{5}}}$

Step 3.2

Multiply $\frac{\sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{2}{3}}{\frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{2}{3}}{\frac{\sqrt{3} \sqrt{5}}{\sqrt{5} \sqrt{5}}}$

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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

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

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

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

Step 3.3.6.4.2

Rewrite the expression.

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

Step 3.3.6.5

Evaluate the exponent.

$\frac{\frac{2}{3}}{\frac{\sqrt{3} \sqrt{5}}{5}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Combine using the product rule for radicals.

$\frac{\frac{2}{3}}{\frac{\sqrt{3 \cdot 5}}{5}}$

Step 3.4.2

Multiply $3$ by $5$ .

$\frac{\frac{2}{3}}{\frac{\sqrt{15}}{5}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{2}{3} \cdot \frac{5}{\sqrt{15}}$

Step 5

Multiply $\frac{5}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{2}{3} (\frac{5}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{5}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 6.2

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{{\sqrt{15}}^{1} \sqrt{15}}$

Step 6.3

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{{\sqrt{15}}^{1} {\sqrt{15}}^{1}}$

Step 6.4

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

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{{\sqrt{15}}^{2}}$

Step 6.6

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 6.6.1

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

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{15^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{15^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{15^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{15}$

Step 7

Cancel the common factor of $5$ and $15$ .

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Step 7.1

Factor $5$ out of $5 \sqrt{15}$ .

$\frac{2}{3} \cdot \frac{5 (\sqrt{15})}{15}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $5$ out of $15$ .

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{5 \cdot 3}$

Step 7.2.2

Cancel the common factor.

$\frac{2}{3} \cdot \frac{5 \sqrt{15}}{5 \cdot 3}$

Step 7.2.3

Rewrite the expression.

$\frac{2}{3} \cdot \frac{\sqrt{15}}{3}$

Step 8

Multiply $\frac{2}{3} \cdot \frac{\sqrt{15}}{3}$ .

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Step 8.1

Multiply $\frac{2}{3}$ by $\frac{\sqrt{15}}{3}$ .

$\frac{2 \sqrt{15}}{3 \cdot 3}$

Step 8.2

Multiply $3$ by $3$ .

$\frac{2 \sqrt{15}}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{15}}{9}$

Decimal Form:

$0.86066296 \dots$