Evaluate ( square root of 75)/2+(3 square root of 2)/4

$\frac{\sqrt{75}}{2} + \frac{3 \sqrt{2}}{4}$

Simplify the numerator.

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$\frac{\sqrt{25 (3)}}{2} + \frac{3 \sqrt{2}}{4}$

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical.

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

To write $\frac{5 \sqrt{3}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Multiply $\frac{5 \sqrt{3}}{2}$ by $\frac{2}{2}$ .

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Multiply $2$ by $5$ .

$\frac{10 \sqrt{3} + 3 \sqrt{2}}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{10 \sqrt{3} + 3 \sqrt{2}}{4}$

Decimal Form:

$5.39078719 \dots$