The value of `sqrt18 + sqrt50 - sqrt32` is

Options

A. `4 sqrt2`
B. `3 sqrt2`
C. `2sqrt2`
D. `sqrt2`

Answer

To solve the expression \( \sqrt{18} + \sqrt{50} - \sqrt{32} \), we will simplify each square root term step by step.

### Step 1: Simplify \( \sqrt{18} \)
- The prime factorization of \( 18 \) is \( 2 \times 3^2 \).
- Therefore, we can write:
\[ \sqrt{18} = \sqrt{2 \times 3^2} = \sqrt{2} \times \sqrt{3^2} = \sqrt{2} \times 3 = 3\sqrt{2} \]

### Step 2: Simplify \( \sqrt{50} \)
- The prime factorization of \( 50 \) is \( 2 \times 5^2 \).
- Thus, we have:
\[ \sqrt{50} = \sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2} = \sqrt{2} \times 5 = 5\sqrt{2} \]

### Step 3: Simplify \( \sqrt{32} \)
- The prime factorization of \( 32 \) is \( 2^5 \).
- We can express this as:
\[ \sqrt{32} = \sqrt{2^5} = \sqrt{(2^4) \times 2} = \sqrt{2^4} \times \sqrt{2} = 4\sqrt{2} \]

### Step 4: Substitute back into the expression
Now, we substitute the simplified square roots back into the original expression:
\[ \sqrt{18} + \sqrt{50} - \sqrt{32} = 3\sqrt{2} + 5\sqrt{2} - 4\sqrt{2} \]

### Step 5: Combine like terms
Combine the terms:
\[ (3 + 5 - 4)\sqrt{2} = 4\sqrt{2} \]

### Final Answer
The value of \( \sqrt{18} + \sqrt{50} - \sqrt{32} \) is:
\[ \boxed{4\sqrt{2}} \]