Trigonometry Examples

$\sqrt{\frac{1 - (- \frac{5 \sqrt{61}}{61})}{1 - \frac{5 \sqrt{61}}{61}}}$

Step 1

Multiply $- (- \frac{5 \sqrt{61}}{61})$ .

Tap for more steps...

Step 1.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{1 + 1 \frac{5 \sqrt{61}}{61}}{1 - \frac{5 \sqrt{61}}{61}}}$

Step 1.2

Multiply $\frac{5 \sqrt{61}}{61}$ by $1$ .

$\sqrt{\frac{1 + \frac{5 \sqrt{61}}{61}}{1 - \frac{5 \sqrt{61}}{61}}}$

Step 2

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

$\sqrt{\frac{\frac{61}{61} + \frac{5 \sqrt{61}}{61}}{1 - \frac{5 \sqrt{61}}{61}}}$

Step 3

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{61 + 5 \sqrt{61}}{61}}{1 - \frac{5 \sqrt{61}}{61}}}$

Step 4

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

$\sqrt{\frac{\frac{61 + 5 \sqrt{61}}{61}}{\frac{61}{61} - \frac{5 \sqrt{61}}{61}}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{61 + 5 \sqrt{61}}{61}}{\frac{61 - 5 \sqrt{61}}{61}}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{61 + 5 \sqrt{61}}{61} \cdot \frac{61}{61 - 5 \sqrt{61}}}$

Step 7

Cancel the common factor of $61$ .

Tap for more steps...

Step 7.1

Cancel the common factor.

$\sqrt{\frac{61 + 5 \sqrt{61}}{61} \cdot \frac{61}{61 - 5 \sqrt{61}}}$

Step 7.2

Rewrite the expression.

$\sqrt{(61 + 5 \sqrt{61}) \frac{1}{61 - 5 \sqrt{61}}}$

Step 8

Multiply $\frac{1}{61 - 5 \sqrt{61}}$ by $\frac{61 + 5 \sqrt{61}}{61 + 5 \sqrt{61}}$ .

$\sqrt{(61 + 5 \sqrt{61}) (\frac{1}{61 - 5 \sqrt{61}} \cdot \frac{61 + 5 \sqrt{61}}{61 + 5 \sqrt{61}})}$

Step 9

Multiply $\frac{1}{61 - 5 \sqrt{61}}$ by $\frac{61 + 5 \sqrt{61}}{61 + 5 \sqrt{61}}$ .

$\sqrt{(61 + 5 \sqrt{61}) \frac{61 + 5 \sqrt{61}}{(61 - 5 \sqrt{61}) (61 + 5 \sqrt{61})}}$

Step 10

Expand the denominator using the FOIL method.

$\sqrt{(61 + 5 \sqrt{61}) \frac{61 + 5 \sqrt{61}}{3721 + 305 \sqrt{61} - 305 \sqrt{61} - 25 {\sqrt{61}}^{2}}}$

Step 11

Simplify.

$\sqrt{(61 + 5 \sqrt{61}) \frac{61 + 5 \sqrt{61}}{2196}}$

Step 12

Apply the distributive property.

$\sqrt{61 \frac{61 + 5 \sqrt{61}}{2196} + 5 \sqrt{61} \frac{61 + 5 \sqrt{61}}{2196}}$

Step 13

Cancel the common factor of $61$ .

Tap for more steps...

Step 13.1

Factor $61$ out of $2196$ .

$\sqrt{61 \frac{61 + 5 \sqrt{61}}{61 (36)} + 5 \sqrt{61} \frac{61 + 5 \sqrt{61}}{2196}}$

Step 13.2

Cancel the common factor.

$\sqrt{61 \frac{61 + 5 \sqrt{61}}{61 \cdot 36} + 5 \sqrt{61} \frac{61 + 5 \sqrt{61}}{2196}}$

Step 13.3

Rewrite the expression.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + 5 \sqrt{61} \frac{61 + 5 \sqrt{61}}{2196}}$

Step 14

Multiply $5 \sqrt{61} \frac{61 + 5 \sqrt{61}}{2196}$ .

Tap for more steps...

Step 14.1

Combine $\frac{61 + 5 \sqrt{61}}{2196}$ and $5$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 + 5 \sqrt{61}) \cdot 5}{2196} \sqrt{61}}$

Step 14.2

Combine $\frac{(61 + 5 \sqrt{61}) \cdot 5}{2196}$ and $\sqrt{61}$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 + 5 \sqrt{61}) \cdot 5 \sqrt{61}}{2196}}$

Step 15

Simplify each term.

Tap for more steps...

Step 15.1

Group $\sqrt{61}$ and $61 + 5 \sqrt{61}$ together.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{\sqrt{61} (61 + 5 \sqrt{61}) \cdot 5}{2196}}$

Step 15.2

Apply the distributive property.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(\sqrt{61} \cdot 61 + \sqrt{61} (5 \sqrt{61})) \cdot 5}{2196}}$

Step 15.3

Move $61$ to the left of $\sqrt{61}$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \cdot \sqrt{61} + \sqrt{61} (5 \sqrt{61})) \cdot 5}{2196}}$

Step 15.4

Multiply $\sqrt{61} (5 \sqrt{61})$ .

Tap for more steps...

Step 15.4.1

Raise $\sqrt{61}$ to the power of $1$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \cdot \sqrt{61} + 5 ({\sqrt{61}}^{1} \sqrt{61})) \cdot 5}{2196}}$

Step 15.4.2

Raise $\sqrt{61}$ to the power of $1$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \cdot \sqrt{61} + 5 ({\sqrt{61}}^{1} {\sqrt{61}}^{1})) \cdot 5}{2196}}$

Step 15.4.3

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

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \cdot \sqrt{61} + 5 {\sqrt{61}}^{1 + 1}) \cdot 5}{2196}}$

Step 15.4.4

Add $1$ and $1$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \cdot \sqrt{61} + 5 {\sqrt{61}}^{2}) \cdot 5}{2196}}$

Step 15.5

Simplify each term.

Tap for more steps...

Step 15.5.1

Rewrite ${\sqrt{61}}^{2}$ as $61$ .

Tap for more steps...

Step 15.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{61}$ as $61^{\frac{1}{2}}$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \sqrt{61} + 5 {(61^{\frac{1}{2}})}^{2}) \cdot 5}{2196}}$

Step 15.5.1.2

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

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \sqrt{61} + 5 \cdot 61^{\frac{1}{2} \cdot 2}) \cdot 5}{2196}}$

Step 15.5.1.3

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

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \sqrt{61} + 5 \cdot 61^{\frac{2}{2}}) \cdot 5}{2196}}$

Step 15.5.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.5.1.4.1

Cancel the common factor.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \sqrt{61} + 5 \cdot 61^{\frac{2}{2}}) \cdot 5}{2196}}$

Step 15.5.1.4.2

Rewrite the expression.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \sqrt{61} + 5 \cdot 61^{1}) \cdot 5}{2196}}$

Step 15.5.1.5

Evaluate the exponent.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \sqrt{61} + 5 \cdot 61) \cdot 5}{2196}}$

Step 15.5.2

Multiply $5$ by $61$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(61 \sqrt{61} + 305) \cdot 5}{2196}}$

Step 15.6

Cancel the common factor of $61 \sqrt{61} + 305$ and $2196$ .

Tap for more steps...

Step 15.6.1

Factor $61$ out of $(61 \sqrt{61} + 305) \cdot 5$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{61 ((\sqrt{61} + 5) \cdot 5)}{2196}}$

Step 15.6.2

Cancel the common factors.

Tap for more steps...

Step 15.6.2.1

Factor $61$ out of $2196$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{61 ((\sqrt{61} + 5) \cdot 5)}{61 (36)}}$

Step 15.6.2.2

Cancel the common factor.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{61 ((\sqrt{61} + 5) \cdot 5)}{61 \cdot 36}}$

Step 15.6.2.3

Rewrite the expression.

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{(\sqrt{61} + 5) \cdot 5}{36}}$

Step 15.7

Move $5$ to the left of $\sqrt{61} + 5$ .

$\sqrt{\frac{61 + 5 \sqrt{61}}{36} + \frac{5 (\sqrt{61} + 5)}{36}}$

Step 16

Combine the numerators over the common denominator.

$\sqrt{\frac{61 + 5 \sqrt{61} + 5 (\sqrt{61} + 5)}{36}}$

Step 17

Simplify each term.

Tap for more steps...

Step 17.1

Apply the distributive property.

$\sqrt{\frac{61 + 5 \sqrt{61} + 5 \sqrt{61} + 5 \cdot 5}{36}}$

Step 17.2

Multiply $5$ by $5$ .

$\sqrt{\frac{61 + 5 \sqrt{61} + 5 \sqrt{61} + 25}{36}}$

Step 18

Simplify terms.

Tap for more steps...

Step 18.1

Add $61$ and $25$ .

$\sqrt{\frac{86 + 5 \sqrt{61} + 5 \sqrt{61}}{36}}$

Step 18.2

Add $5 \sqrt{61}$ and $5 \sqrt{61}$ .

$\sqrt{\frac{86 + 10 \sqrt{61}}{36}}$

Step 18.3

Cancel the common factor of $86 + 10 \sqrt{61}$ and $36$ .

Tap for more steps...

Step 18.3.1

Factor $2$ out of $86$ .

$\sqrt{\frac{2 (43) + 10 \sqrt{61}}{36}}$

Step 18.3.2

Factor $2$ out of $10 \sqrt{61}$ .

$\sqrt{\frac{2 (43) + 2 (5 \sqrt{61})}{36}}$

Step 18.3.3

Factor $2$ out of $2 (43) + 2 (5 \sqrt{61})$ .

$\sqrt{\frac{2 (43 + 5 \sqrt{61})}{36}}$

Step 18.3.4

Cancel the common factors.

Tap for more steps...

Step 18.3.4.1

Factor $2$ out of $36$ .

$\sqrt{\frac{2 (43 + 5 \sqrt{61})}{2 \cdot 18}}$

Step 18.3.4.2

Cancel the common factor.

$\sqrt{\frac{2 (43 + 5 \sqrt{61})}{2 \cdot 18}}$

Step 18.3.4.3

Rewrite the expression.

$\sqrt{\frac{43 + 5 \sqrt{61}}{18}}$

Step 19

Rewrite $\sqrt{\frac{43 + 5 \sqrt{61}}{18}}$ as $\frac{\sqrt{43 + 5 \sqrt{61}}}{\sqrt{18}}$ .

$\frac{\sqrt{43 + 5 \sqrt{61}}}{\sqrt{18}}$

Step 20

Simplify the denominator.

Tap for more steps...

Step 20.1

Rewrite $18$ as $3^{2} \cdot 2$ .

Tap for more steps...

Step 20.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{43 + 5 \sqrt{61}}}{\sqrt{9 (2)}}$

Step 20.1.2

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

$\frac{\sqrt{43 + 5 \sqrt{61}}}{\sqrt{3^{2} \cdot 2}}$

Step 20.2

Pull terms out from under the radical.

$\frac{\sqrt{43 + 5 \sqrt{61}}}{3 \sqrt{2}}$

Step 21

Multiply $\frac{\sqrt{43 + 5 \sqrt{61}}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{43 + 5 \sqrt{61}}}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 22

Combine and simplify the denominator.

Tap for more steps...

Step 22.1

Multiply $\frac{\sqrt{43 + 5 \sqrt{61}}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 \sqrt{2} \sqrt{2}}$

Step 22.2

Move $\sqrt{2}$ .

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 (\sqrt{2} \sqrt{2})}$

Step 22.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 22.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 22.5

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

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 22.6

Add $1$ and $1$ .

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 {\sqrt{2}}^{2}}$

Step 22.7

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

Tap for more steps...

Step 22.7.1

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

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

Step 22.7.2

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

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 22.7.3

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

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 22.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.7.4.1

Cancel the common factor.

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 22.7.4.2

Rewrite the expression.

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 \cdot 2^{1}}$

Step 22.7.5

Evaluate the exponent.

$\frac{\sqrt{43 + 5 \sqrt{61}} \sqrt{2}}{3 \cdot 2}$

Step 23

Combine using the product rule for radicals.

$\frac{\sqrt{(43 + 5 \sqrt{61}) \cdot 2}}{3 \cdot 2}$

Step 24

Multiply $3$ by $2$ .

$\frac{\sqrt{(43 + 5 \sqrt{61}) \cdot 2}}{6}$

Step 25

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(43 + 5 \sqrt{61}) \cdot 2}}{6}$

Decimal Form:

$2.13504161 \dots$