If the value of `sqrt5 = 2.236` then calculate the value of `sqrt(405) - 1/2 sqrt(80)- sqrt(125)`

Options

A. 2.236
B. 4.472
C. 1.118
D. 8.994

Answer

To solve the expression \( \sqrt{405} - \frac{1}{2} \sqrt{80} - \sqrt{125} \) given that \( \sqrt{5} = 2.236 \), we can follow these steps:

### Step 1: Simplify each square root
1. **Calculate \( \sqrt{405} \)**:
- We can factor \( 405 \) as \( 81 \times 5 \).
- Therefore, \( \sqrt{405} = \sqrt{81 \times 5} = \sqrt{81} \times \sqrt{5} = 9\sqrt{5} \).

2. **Calculate \( \sqrt{80} \)**:
- We can factor \( 80 \) as \( 16 \times 5 \).
- Therefore, \( \sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5} \).

3. **Calculate \( \sqrt{125} \)**:
- We can factor \( 125 \) as \( 25 \times 5 \).
- Therefore, \( \sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5} \).

### Step 2: Substitute back into the expression
Now we substitute these values back into the original expression:
\[ \sqrt{405} - \frac{1}{2} \sqrt{80} - \sqrt{125} = 9\sqrt{5} - \frac{1}{2}(4\sqrt{5}) - 5\sqrt{5} \]

### Step 3: Simplify the expression
1. **Calculate \( \frac{1}{2}(4\sqrt{5}) \)**:
- This simplifies to \( 2\sqrt{5} \).

2. **Now substitute this back**:
\[ 9\sqrt{5} - 2\sqrt{5} - 5\sqrt{5} \]

3. **Combine like terms**:
\[ (9\sqrt{5} - 2\sqrt{5} - 5\sqrt{5}) = (9 - 2 - 5)\sqrt{5} = 2\sqrt{5} \]

### Step 4: Substitute the value of \( \sqrt{5} \)
Now we substitute the value of \( \sqrt{5} = 2.236 \):
\[ 2\sqrt{5} = 2 \times 2.236 = 4.472 \]

### Final Answer
Thus, the value of the expression \( \sqrt{405} - \frac{1}{2} \sqrt{80} - \sqrt{125} \) is **4.472**.

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