Solve or simplify the following problem, using the properties of roots: `sqrt(150) - sqrt(96)`

Options

A. `sqrt(4)`
B. `sqrt(6)`
C. `sqrt(5)`
D. `sqrt(3)`

Answer

To solve the expression \( \sqrt{150} - \sqrt{96} \), we will simplify each square root using the properties of roots.

### Step 1: Simplify \( \sqrt{150} \)

First, we can factor \( 150 \):
\[ 150 = 15 \times 10 \]
Next, we can break down \( 15 \) and \( 10 \):
\[ 15 = 3 \times 5 \quad \text{and} \quad 10 = 2 \times 5 \]
Thus, we can rewrite \( 150 \) as:
\[ 150 = 3 \times 5 \times 2 \times 5 = 5^2 \times 6 \]
Now, we can take the square root:
\[ \sqrt{150} = \sqrt{5^2 \times 6} = 5\sqrt{6} \]

### Step 2: Simplify \( \sqrt{96} \)

Now, let's simplify \( \sqrt{96} \):
\[ 96 = 16 \times 6 \]
We know that \( 16 \) is a perfect square:
\[ 16 = 4^2 \]
Thus, we can rewrite \( 96 \) as:
\[ 96 = 4^2 \times 6 \]
Now, we can take the square root:
\[ \sqrt{96} = \sqrt{4^2 \times 6} = 4\sqrt{6} \]

### Step 3: Subtract the two square roots

Now we can substitute the simplified forms back into the original expression:
\[ \sqrt{150} - \sqrt{96} = 5\sqrt{6} - 4\sqrt{6} \]
Since both terms have a common factor of \( \sqrt{6} \), we can factor it out:
\[ = (5 - 4)\sqrt{6} = 1\sqrt{6} = \sqrt{6} \]

### Final Answer

Thus, the simplified form of \( \sqrt{150} - \sqrt{96} \) is:
\[ \sqrt{6} \]

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