Trigonometry Examples

$\sqrt{\frac{\sqrt{13} - 3}{\sqrt{13}} \cdot 2}$

Step 1

Combine $\frac{\sqrt{13} - 3}{\sqrt{13}}$ and $2$ .

$\sqrt{\frac{(\sqrt{13} - 3) \cdot 2}{\sqrt{13}}}$

Step 2

Move $2$ to the left of $\sqrt{13} - 3$ .

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3)}{\sqrt{13}}}$

Step 3

Multiply $\frac{2 \cdot (\sqrt{13} - 3)}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3)}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}}$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Combine and simplify the denominator.

Tap for more steps...

Step 4.1.1

Multiply $\frac{2 \cdot (\sqrt{13} - 3)}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{\sqrt{13} \sqrt{13}}}$

Step 4.1.2

Raise $\sqrt{13}$ to the power of $1$ .

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}}$

Step 4.1.3

Raise $\sqrt{13}$ to the power of $1$ .

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}}$

Step 4.1.4

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

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{{\sqrt{13}}^{1 + 1}}}$

Step 4.1.5

Add $1$ and $1$ .

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{{\sqrt{13}}^{2}}}$

Step 4.1.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Tap for more steps...

Step 4.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}}$

Step 4.1.6.2

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

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}}$

Step 4.1.6.3

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

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{13^{\frac{2}{2}}}}$

Step 4.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.6.4.1

Cancel the common factor.

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{13^{\frac{2}{2}}}}$

Step 4.1.6.4.2

Rewrite the expression.

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{13^{1}}}$

Step 4.1.6.5

Evaluate the exponent.

$\sqrt{\frac{2 \cdot (\sqrt{13} - 3) \sqrt{13}}{13}}$

Step 4.2

Group $\sqrt{13}$ and $\sqrt{13} - 3$ together.

$\sqrt{\frac{2 \cdot (\sqrt{13} (\sqrt{13} - 3))}{13}}$

Step 4.3

Apply the distributive property.

$\sqrt{\frac{2 \cdot (\sqrt{13} \sqrt{13} + \sqrt{13} \cdot - 3)}{13}}$

Step 4.4

Combine using the product rule for radicals.

$\sqrt{\frac{2 \cdot (\sqrt{13 \cdot 13} + \sqrt{13} \cdot - 3)}{13}}$

Step 4.5

Move $- 3$ to the left of $\sqrt{13}$ .

$\sqrt{\frac{2 \cdot (\sqrt{13 \cdot 13} - 3 \cdot \sqrt{13})}{13}}$

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Multiply $13$ by $13$ .

$\sqrt{\frac{2 \cdot (\sqrt{169} - 3 \cdot \sqrt{13})}{13}}$

Step 5.2

Rewrite $169$ as $13^{2}$ .

$\sqrt{\frac{2 \cdot (\sqrt{13^{2}} - 3 \cdot \sqrt{13})}{13}}$

Step 5.3

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

$\sqrt{\frac{2 \cdot (13 - 3 \sqrt{13})}{13}}$

Step 6

Rewrite $\sqrt{\frac{2 (13 - 3 \sqrt{13})}{13}}$ as $\frac{\sqrt{2 (13 - 3 \sqrt{13})}}{\sqrt{13}}$ .

$\frac{\sqrt{2 (13 - 3 \sqrt{13})}}{\sqrt{13}}$

Step 7

Multiply $\frac{\sqrt{2 (13 - 3 \sqrt{13})}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{\sqrt{2 (13 - 3 \sqrt{13})}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Step 8

Combine and simplify the denominator.

Tap for more steps...

Step 8.1

Multiply $\frac{\sqrt{2 (13 - 3 \sqrt{13})}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 8.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}$

Step 8.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}$

Step 8.4

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

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{{\sqrt{13}}^{2}}$

Step 8.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Tap for more steps...

Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{13^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{13^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{13^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt{2 (13 - 3 \sqrt{13})} \sqrt{13}}{13}$

Step 9

Simplify the numerator.

Tap for more steps...

Step 9.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 (13 - 3 \sqrt{13}) \cdot 13}}{13}$

Step 9.2

Multiply $13$ by $2$ .

$\frac{\sqrt{26 (13 - 3 \sqrt{13})}}{13}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{26 (13 - 3 \sqrt{13})}}{13}$

Decimal Form:

$0.57956829 \dots$