Trigonometry Examples

$\frac{\sqrt{33}}{\sqrt{55}}$

Step 1

Combine $\sqrt{33}$ and $\sqrt{55}$ into a single radical.

$\sqrt{\frac{33}{55}}$

Step 2

Cancel the common factor of $33$ and $55$ .

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Step 2.1

Factor $11$ out of $33$ .

$\sqrt{\frac{11 (3)}{55}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $11$ out of $55$ .

$\sqrt{\frac{11 \cdot 3}{11 \cdot 5}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{11 \cdot 3}{11 \cdot 5}}$

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

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

Step 5.6.5

Evaluate the exponent.

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

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $3$ by $5$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.77459666 \dots$