Trigonometry Examples

$\frac{\sqrt{7}}{\sqrt{8 k}}$

Step 1

Simplify the denominator.

Tap for more steps...

Step 1.1

Rewrite $8 k$ as $2^{2} \cdot (2 k)$ .

Tap for more steps...

Step 1.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{7}}{\sqrt{4 (2) k}}$

Step 1.1.2

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

$\frac{\sqrt{7}}{\sqrt{2^{2} \cdot 2 k}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt{7}}{\sqrt{2^{2} \cdot (2 k)}}$

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{\sqrt{7}}{2 \sqrt{2 k}}$ by $\frac{\sqrt{2 k}}{\sqrt{2 k}}$ .

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

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

Multiply $\frac{\sqrt{7}}{2 \sqrt{2 k}}$ by $\frac{\sqrt{2 k}}{\sqrt{2 k}}$ .

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

Step 3.2

Move $\sqrt{2 k}$ .

$\frac{\sqrt{7} \sqrt{2 k}}{2 (\sqrt{2 k} \sqrt{2 k})}$

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$\frac{\sqrt{7} \sqrt{2 k}}{2 {\sqrt{2 k}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$\frac{\sqrt{7} \sqrt{2 k}}{2 {\sqrt{2 k}}^{2}}$

Step 3.7

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

Tap for more steps...

Step 3.7.1

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

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

Step 3.7.2

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

$\frac{\sqrt{7} \sqrt{2 k}}{2 {(2 k)}^{\frac{1}{2} \cdot 2}}$

Step 3.7.3

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

$\frac{\sqrt{7} \sqrt{2 k}}{2 {(2 k)}^{\frac{2}{2}}}$

Step 3.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.7.4.1

Cancel the common factor.

$\frac{\sqrt{7} \sqrt{2 k}}{2 {(2 k)}^{\frac{2}{2}}}$

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Simplify.

$\frac{\sqrt{7} \sqrt{2 k}}{2 (2 k)}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt{7 (2 k)}}{2 \cdot 2 k}$

Step 4.2

Multiply $7$ by $2$ .

$\frac{\sqrt{14 k}}{2 \cdot 2 k}$

Step 5

Multiply $2$ by $2$ .

$\frac{\sqrt{14 k}}{4 k}$