Trigonometry Examples

$\sqrt{\frac{1 - \frac{\sqrt{63}}{21}}{2}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $63$ as $3^{2} \cdot 7$ .

Tap for more steps...

Step 1.1.1

Factor $9$ out of $63$ .

$\sqrt{\frac{1 - \frac{\sqrt{9 (7)}}{21}}{2}}$

Step 1.1.2

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

$\sqrt{\frac{1 - \frac{\sqrt{3^{2} \cdot 7}}{21}}{2}}$

Step 1.2

Pull terms out from under the radical.

$\sqrt{\frac{1 - \frac{3 \sqrt{7}}{21}}{2}}$

Step 2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.1

Cancel the common factor of $3$ and $21$ .

Tap for more steps...

Step 2.1.1

Factor $3$ out of $3 \sqrt{7}$ .

$\sqrt{\frac{1 - \frac{3 (\sqrt{7})}{21}}{2}}$

Step 2.1.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.1

Factor $3$ out of $21$ .

$\sqrt{\frac{1 - \frac{3 \sqrt{7}}{3 \cdot 7}}{2}}$

Step 2.1.2.2

Cancel the common factor.

$\sqrt{\frac{1 - \frac{3 \sqrt{7}}{3 \cdot 7}}{2}}$

Step 2.1.2.3

Rewrite the expression.

$\sqrt{\frac{1 - \frac{\sqrt{7}}{7}}{2}}$

Step 2.2

Simplify the expression.

Tap for more steps...

Step 2.2.1

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

$\sqrt{\frac{\frac{7}{7} - \frac{\sqrt{7}}{7}}{2}}$

Step 2.2.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{7 - \sqrt{7}}{7}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{7 - \sqrt{7}}{7} \cdot \frac{1}{2}}$

Step 4

Multiply $\frac{7 - \sqrt{7}}{7} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 4.1

Multiply $\frac{7 - \sqrt{7}}{7}$ by $\frac{1}{2}$ .

$\sqrt{\frac{7 - \sqrt{7}}{7 \cdot 2}}$

Step 4.2

Multiply $7$ by $2$ .

$\sqrt{\frac{7 - \sqrt{7}}{14}}$

Step 5

Rewrite $\sqrt{\frac{7 - \sqrt{7}}{14}}$ as $\frac{\sqrt{7 - \sqrt{7}}}{\sqrt{14}}$ .

$\frac{\sqrt{7 - \sqrt{7}}}{\sqrt{14}}$

Step 6

Multiply $\frac{\sqrt{7 - \sqrt{7}}}{\sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$\frac{\sqrt{7 - \sqrt{7}}}{\sqrt{14}} \cdot \frac{\sqrt{14}}{\sqrt{14}}$

Step 7

Combine and simplify the denominator.

Tap for more steps...

Step 7.1

Multiply $\frac{\sqrt{7 - \sqrt{7}}}{\sqrt{14}}$ by $\frac{\sqrt{14}}{\sqrt{14}}$ .

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{\sqrt{14} \sqrt{14}}$

Step 7.2

Raise $\sqrt{14}$ to the power of $1$ .

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{{\sqrt{14}}^{1} \sqrt{14}}$

Step 7.3

Raise $\sqrt{14}$ to the power of $1$ .

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{{\sqrt{14}}^{1} {\sqrt{14}}^{1}}$

Step 7.4

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

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{{\sqrt{14}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{{\sqrt{14}}^{2}}$

Step 7.6

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Tap for more steps...

Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{{(14^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{14^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{14^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{14^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{14^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{7 - \sqrt{7}} \sqrt{14}}{14}$

Step 8

Combine using the product rule for radicals.

$\frac{\sqrt{(7 - \sqrt{7}) \cdot 14}}{14}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(7 - \sqrt{7}) \cdot 14}}{14}$

Decimal Form:

$0.55768966 \dots$