Trigonometry Examples

$\frac{\sqrt{20 y}}{y \sqrt{5 + 1}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $20 y$ as $2^{2} \cdot (5 y)$ .

Tap for more steps...

Step 1.1.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5) y}}{y \sqrt{5 + 1}}$

Step 1.1.2

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

$\frac{\sqrt{2^{2} \cdot 5 y}}{y \sqrt{5 + 1}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt{2^{2} \cdot (5 y)}}{y \sqrt{5 + 1}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{5 y}}{y \sqrt{5 + 1}}$

Step 2

Add $5$ and $1$ .

$\frac{2 \sqrt{5 y}}{y \sqrt{6}}$

Step 3

Multiply $\frac{2 \sqrt{5 y}}{y \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2 \sqrt{5 y}}{y \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Combine and simplify the denominator.

Tap for more steps...

Step 4.1.1

Multiply $\frac{2 \sqrt{5 y}}{y \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2 \sqrt{5 y} \sqrt{6}}{y \sqrt{6} \sqrt{6}}$

Step 4.1.2

Move $\sqrt{6}$ .

$\frac{2 \sqrt{5 y} \sqrt{6}}{y (\sqrt{6} \sqrt{6})}$

Step 4.1.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{5 y} \sqrt{6}}{y ({\sqrt{6}}^{1} \sqrt{6})}$

Step 4.1.4

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{5 y} \sqrt{6}}{y ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Step 4.1.5

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

$\frac{2 \sqrt{5 y} \sqrt{6}}{y {\sqrt{6}}^{1 + 1}}$

Step 4.1.6

Add $1$ and $1$ .

$\frac{2 \sqrt{5 y} \sqrt{6}}{y {\sqrt{6}}^{2}}$

Step 4.1.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 4.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{2 \sqrt{5 y} \sqrt{6}}{y {(6^{\frac{1}{2}})}^{2}}$

Step 4.1.7.2

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

$\frac{2 \sqrt{5 y} \sqrt{6}}{y \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 4.1.7.3

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

$\frac{2 \sqrt{5 y} \sqrt{6}}{y \cdot 6^{\frac{2}{2}}}$

Step 4.1.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.7.4.1

Cancel the common factor.

$\frac{2 \sqrt{5 y} \sqrt{6}}{y \cdot 6^{\frac{2}{2}}}$

Step 4.1.7.4.2

Rewrite the expression.

$\frac{2 \sqrt{5 y} \sqrt{6}}{y \cdot 6^{1}}$

Step 4.1.7.5

Evaluate the exponent.

$\frac{2 \sqrt{5 y} \sqrt{6}}{y \cdot 6}$

Step 4.2

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 4.2.1

Factor $2$ out of $2 \sqrt{5 y} \sqrt{6}$ .

$\frac{2 (\sqrt{5 y} \sqrt{6})}{y \cdot 6}$

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

Factor $2$ out of $y \cdot 6$ .

$\frac{2 (\sqrt{5 y} \sqrt{6})}{2 (y \cdot 3)}$

Step 4.2.2.2

Cancel the common factor.

$\frac{2 (\sqrt{5 y} \sqrt{6})}{2 (y \cdot 3)}$

Step 4.2.2.3

Rewrite the expression.

$\frac{\sqrt{5 y} \sqrt{6}}{y \cdot 3}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 y \cdot 6}}{y \cdot 3}$

Step 5.2

Multiply $6$ by $5$ .

$\frac{\sqrt{30 y}}{y \cdot 3}$

Step 6

Move $3$ to the left of $y$ .

$\frac{\sqrt{30 y}}{3 y}$