The problem is to simplify the expression $\sqrt{75} - \sqrt{192}$.

1. Problem Description

The problem is to simplify the expression $\sqrt{75} - \sqrt{192}$.

2. Solution Steps

First, we simplify $\sqrt{75}$. We look for the largest perfect square that divides

7

5. $75 = 25 \times 3 = 5^2 \times 3$.

Therefore, $\sqrt{75} = \sqrt{5^2 \times 3} = \sqrt{5^2} \times \sqrt{3} = 5\sqrt{3}$.

Next, we simplify $\sqrt{192}$. We look for the largest perfect square that divides

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9

2. We can start by dividing 192 by 4: $192 = 4 \times 48$.

Then $48 = 16 \times 3 = 4^2 \times 3$.

So $192 = 4 \times 16 \times 3 = 64 \times 3 = 8^2 \times 3$.

Therefore, $\sqrt{192} = \sqrt{8^2 \times 3} = \sqrt{8^2} \times \sqrt{3} = 8\sqrt{3}$.

Now we can substitute the simplified expressions back into the original expression:

$\sqrt{75} - \sqrt{192} = 5\sqrt{3} - 8\sqrt{3}$.

Since we have like terms, we can combine them:

$5\sqrt{3} - 8\sqrt{3} = (5 - 8)\sqrt{3} = -3\sqrt{3}$.

3. Final Answer

$-3\sqrt{3}$