Trigonometry Examples

$\frac{3 \sqrt{8}}{3 \sqrt{12}}$

Step 1

Cancel the common factor of $3$ .

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Step 1.1

Cancel the common factor.

$\frac{3 \sqrt{8}}{3 \sqrt{12}}$

Step 1.2

Rewrite the expression.

$\frac{\sqrt{8}}{\sqrt{12}}$

Step 2

Combine $\sqrt{8}$ and $\sqrt{12}$ into a single radical.

$\sqrt{\frac{8}{12}}$

Step 3

Cancel the common factor of $8$ and $12$ .

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Step 3.1

Factor $4$ out of $8$ .

$\sqrt{\frac{4 (2)}{12}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4$ out of $12$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

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

Step 6.6.5

Evaluate the exponent.

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

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{3}$

Decimal Form:

$0.81649658 \dots$