Basic Math Examples

$\frac{\sqrt{81}}{\sqrt{36}} \div \frac{\sqrt{3}}{\sqrt{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{\sqrt{81}}{\sqrt{36}} \cdot \frac{\sqrt{2}}{\sqrt{3}}$

Step 2

Combine $\sqrt{81}$ and $\sqrt{36}$ into a single radical.

$\sqrt{\frac{81}{36}} \frac{\sqrt{2}}{\sqrt{3}}$

Step 3

Cancel the common factor of $81$ and $36$ .

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Step 3.1

Factor $9$ out of $81$ .

$\sqrt{\frac{9 (9)}{36}} \frac{\sqrt{2}}{\sqrt{3}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $9$ out of $36$ .

$\sqrt{\frac{9 \cdot 9}{9 \cdot 4}} \frac{\sqrt{2}}{\sqrt{3}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{9 \cdot 9}{9 \cdot 4}} \frac{\sqrt{2}}{\sqrt{3}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{9}{4}} \frac{\sqrt{2}}{\sqrt{3}}$

Step 4

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

$\frac{\sqrt{9}}{\sqrt{4}} \cdot \frac{\sqrt{2}}{\sqrt{3}}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{\sqrt{3^{2}}}{\sqrt{4}} \cdot \frac{\sqrt{2}}{\sqrt{3}}$

Step 5.2

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

$\frac{3}{\sqrt{4}} \cdot \frac{\sqrt{2}}{\sqrt{3}}$

Step 6

Simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Evaluate the exponent.

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

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

Combine using the product rule for radicals.

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

Step 11

Multiply $2$ by $3$ .

$\frac{1}{2} \sqrt{6}$

Step 12

Combine $\frac{1}{2}$ and $\sqrt{6}$ .

$\frac{\sqrt{6}}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{2}$

Decimal Form:

$1.22474487 \dots$