Trigonometry Examples

$\sqrt{\frac{\frac{2 \sqrt{42}}{16} + 1}{2}}$

Step 1

Simplify the expression.

Tap for more steps...

Step 1.1

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

$\sqrt{\frac{\frac{2 \sqrt{42}}{16} + \frac{16}{16}}{2}}$

Step 1.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 \sqrt{42} + 16}{16}}{2}}$

Step 2

Reduce the expression $\frac{2 \sqrt{42} + 16}{16}$ by cancelling the common factors.

Tap for more steps...

Step 2.1

Factor $2$ out of $2 \sqrt{42}$ .

$\sqrt{\frac{\frac{2 (\sqrt{42}) + 16}{16}}{2}}$

Step 2.2

Factor $2$ out of $16$ .

$\sqrt{\frac{\frac{2 \sqrt{42} + 2 \cdot 8}{16}}{2}}$

Step 2.3

Factor $2$ out of $2 \sqrt{42} + 2 \cdot 8$ .

$\sqrt{\frac{\frac{2 (\sqrt{42} + 8)}{16}}{2}}$

Step 2.4

Factor $2$ out of $16$ .

$\sqrt{\frac{\frac{2 (\sqrt{42} + 8)}{2 (8)}}{2}}$

Step 2.5

Cancel the common factor.

$\sqrt{\frac{\frac{2 (\sqrt{42} + 8)}{2 \cdot 8}}{2}}$

Step 2.6

Rewrite the expression.

$\sqrt{\frac{\frac{\sqrt{42} + 8}{8}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{\sqrt{42} + 8}{8} \cdot \frac{1}{2}}$

Step 4

Multiply $\frac{\sqrt{42} + 8}{8} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 4.1

Multiply $\frac{\sqrt{42} + 8}{8}$ by $\frac{1}{2}$ .

$\sqrt{\frac{\sqrt{42} + 8}{8 \cdot 2}}$

Step 4.2

Multiply $8$ by $2$ .

$\sqrt{\frac{\sqrt{42} + 8}{16}}$

Step 5

Rewrite $\sqrt{\frac{\sqrt{42} + 8}{16}}$ as $\frac{\sqrt{\sqrt{42} + 8}}{\sqrt{16}}$ .

$\frac{\sqrt{\sqrt{42} + 8}}{\sqrt{16}}$

Step 6

Simplify the denominator.

Tap for more steps...

Step 6.1

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{\sqrt{42} + 8}}{\sqrt{4^{2}}}$

Step 6.2

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

$\frac{\sqrt{\sqrt{42} + 8}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{\sqrt{42} + 8}}{4}$

Decimal Form:

$0.95133921 \dots$