Simplifying Square Roots

The problem requires simplifying and adding two square roots: $\sqrt{75}$ and $\sqrt{147}$.

Step 1: Simplify $\sqrt{75}$

Find the largest perfect square that divides 75. This is 25, since $75 = 25 \times 3$. Therefore:

$\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$

Step 2: Simplify $\sqrt{147}$

Find the largest perfect square that divides 147. This is 49, since $147 = 49 \times 3$. Therefore:

$\sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7\sqrt{3}$

Step 3: Add the Simplified Roots

Now add the results from Step 1 and Step 2:

$5\sqrt{3} + 7\sqrt{3}$

Since both terms have the same radical part ($\sqrt{3}$), we can add their coefficients:

$(5 + 7)\sqrt{3} = 12\sqrt{3}$

Final Value Calculation

The value of $\sqrt{75} + \sqrt{147}$ is $12\sqrt{3}$. This corresponds to Option 4.