Simplify ( square root of 45x)/4+ square root of (5x)/4

$\frac{\sqrt{45 x}}{4} + \sqrt{\frac{5 x}{4}}$

Step 1

Simplify each term.

Simplify the numerator.

Rewrite $45 x$ as $3^{2} \cdot (5 x)$ .

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5) x}}{4} + \sqrt{\frac{5 x}{4}}$

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

$\frac{\sqrt{3^{2} \cdot 5 x}}{4} + \sqrt{\frac{5 x}{4}}$

Add parentheses.

$\frac{\sqrt{3^{2} \cdot (5 x)}}{4} + \sqrt{\frac{5 x}{4}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{5 x}}{4} + \sqrt{\frac{5 x}{4}}$

Rewrite $\frac{5 x}{4}$ as ${(\frac{1}{2})}^{2} (5 x)$ .

Factor the perfect power $1^{2}$ out of $5 x$ .

$\frac{3 \sqrt{5 x}}{4} + \sqrt{\frac{1^{2} (5 x)}{4}}$

Factor the perfect power $2^{2}$ out of $4$ .

$\frac{3 \sqrt{5 x}}{4} + \sqrt{\frac{1^{2} (5 x)}{2^{2} \cdot 1}}$

Rearrange the fraction $\frac{1^{2} (5 x)}{2^{2} \cdot 1}$ .

$\frac{3 \sqrt{5 x}}{4} + \sqrt{{(\frac{1}{2})}^{2} (5 x)}$

Pull terms out from under the radical.

$\frac{3 \sqrt{5 x}}{4} + \frac{1}{2} \sqrt{5 x}$

Combine $\frac{1}{2}$ and $\sqrt{5 x}$ .

$\frac{3 \sqrt{5 x}}{4} + \frac{\sqrt{5 x}}{2}$

Step 2

To write $\frac{\sqrt{5 x}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 \sqrt{5 x}}{4} + \frac{\sqrt{5 x}}{2} \cdot \frac{2}{2}$

Step 3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{\sqrt{5 x}}{2}$ by $\frac{2}{2}$ .

$\frac{3 \sqrt{5 x}}{4} + \frac{\sqrt{5 x} \cdot 2}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{3 \sqrt{5 x}}{4} + \frac{\sqrt{5 x} \cdot 2}{4}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 \sqrt{5 x} + \sqrt{5 x} \cdot 2}{4}$

Step 5

Simplify the numerator.

Move $2$ to the left of $\sqrt{5 x}$ .

$\frac{3 \sqrt{5 x} + 2 \cdot \sqrt{5 x}}{4}$

Add $3 \sqrt{5 x}$ and $2 \sqrt{5 x}$ .

$\frac{5 \sqrt{5 x}}{4}$