Trigonometry Examples

$\sqrt{1 - \frac{\frac{8 \sqrt{113}}{113}}{2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1 - (\frac{8 \sqrt{113}}{113} \cdot \frac{1}{2})}$

Step 2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1

Factor $2$ out of $8 \sqrt{113}$ .

$\sqrt{1 - (\frac{2 (4 \sqrt{113})}{113} \cdot \frac{1}{2})}$

Step 2.2

Cancel the common factor.

$\sqrt{1 - (\frac{2 (4 \sqrt{113})}{113} \cdot \frac{1}{2})}$

Step 2.3

Rewrite the expression.

$\sqrt{1 - \frac{4 \sqrt{113}}{113}}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{113}{113} - \frac{4 \sqrt{113}}{113}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{113 - 4 \sqrt{113}}{113}}$

Step 4

Rewrite $\sqrt{\frac{113 - 4 \sqrt{113}}{113}}$ as $\frac{\sqrt{113 - 4 \sqrt{113}}}{\sqrt{113}}$ .

$\frac{\sqrt{113 - 4 \sqrt{113}}}{\sqrt{113}}$

Step 5

Multiply $\frac{\sqrt{113 - 4 \sqrt{113}}}{\sqrt{113}}$ by $\frac{\sqrt{113}}{\sqrt{113}}$ .

$\frac{\sqrt{113 - 4 \sqrt{113}}}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}}$

Step 6

Combine and simplify the denominator.

Tap for more steps...

Step 6.1

Multiply $\frac{\sqrt{113 - 4 \sqrt{113}}}{\sqrt{113}}$ by $\frac{\sqrt{113}}{\sqrt{113}}$ .

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{\sqrt{113} \sqrt{113}}$

Step 6.2

Raise $\sqrt{113}$ to the power of $1$ .

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1} \sqrt{113}}$

Step 6.3

Raise $\sqrt{113}$ to the power of $1$ .

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1} {\sqrt{113}}^{1}}$

Step 6.4

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

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{{\sqrt{113}}^{2}}$

Step 6.6

Rewrite ${\sqrt{113}}^{2}$ as $113$ .

Tap for more steps...

Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{113}$ as $113^{\frac{1}{2}}$ .

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{113^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.6.4.1

Cancel the common factor.

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{113^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{113^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{\sqrt{113 - 4 \sqrt{113}} \sqrt{113}}{113}$

Step 7

Combine using the product rule for radicals.

$\frac{\sqrt{(113 - 4 \sqrt{113}) \cdot 113}}{113}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(113 - 4 \sqrt{113}) \cdot 113}}{113}$

Decimal Form:

$0.78975417 \dots$