Answer :

To simplify the expression

[tex]$$
\sqrt{\frac{45}{12}},
$$[/tex]

follow these steps:

1. Simplify the Fraction:
Divide both the numerator and the denominator by their greatest common divisor, which is [tex]$3$[/tex]. This yields

[tex]$$
\frac{45}{12} = \frac{45 \div 3}{12 \div 3} = \frac{15}{4}.
$$[/tex]

2. Apply the Square Root:
Rewrite the square root of the fraction as the fraction of square roots:

[tex]$$
\sqrt{\frac{15}{4}} = \frac{\sqrt{15}}{\sqrt{4}}.
$$[/tex]

3. Simplify the Denominator:
Since [tex]$\sqrt{4} = 2$[/tex], the expression becomes

[tex]$$
\frac{\sqrt{15}}{2}.
$$[/tex]

Thus, the simplified form of the original expression is

[tex]$$
\boxed{\frac{\sqrt{15}}{2}}.
$$[/tex]