If `165^(2)=27225`, then what is the value of `sqrt(272.25)+sqrt(2.7225)+sqrt(0.027225)` ?

Options

A. `(A)12.25`
B. `(B)34.65`
C. (C)`19.8`
D. (D)`18.315`

Answer

To solve the problem, we need to find the value of \( \sqrt{272.25} + \sqrt{2.7225} + \sqrt{0.027225} \).

### Step-by-Step Solution:

1. **Calculate \( \sqrt{272.25} \)**:
- We can express \( 272.25 \) as \( \frac{27225}{100} \).
- Therefore, \( \sqrt{272.25} = \sqrt{\frac{27225}{100}} = \frac{\sqrt{27225}}{\sqrt{100}} = \frac{\sqrt{27225}}{10} \).
- Since \( 165^2 = 27225 \), we have \( \sqrt{27225} = 165 \).
- Thus, \( \sqrt{272.25} = \frac{165}{10} = 16.5 \).

2. **Calculate \( \sqrt{2.7225} \)**:
- We can express \( 2.7225 \) as \( \frac{27225}{10000} \).
- Therefore, \( \sqrt{2.7225} = \sqrt{\frac{27225}{10000}} = \frac{\sqrt{27225}}{\sqrt{10000}} = \frac{\sqrt{27225}}{100} \).
- Again, since \( \sqrt{27225} = 165 \), we have \( \sqrt{2.7225} = \frac{165}{100} = 1.65 \).

3. **Calculate \( \sqrt{0.027225} \)**:
- We can express \( 0.027225 \) as \( \frac{27225}{1000000} \).
- Therefore, \( \sqrt{0.027225} = \sqrt{\frac{27225}{1000000}} = \frac{\sqrt{27225}}{\sqrt{1000000}} = \frac{\sqrt{27225}}{1000} \).
- Again, since \( \sqrt{27225} = 165 \), we have \( \sqrt{0.027225} = \frac{165}{1000} = 0.165 \).

4. **Add the results together**:
- Now, we add the three square roots we calculated:
\[ \sqrt{272.25} + \sqrt{2.7225} + \sqrt{0.027225} = 16.5 + 1.65 + 0.165 \]
- Performing the addition:
\[ 16.5 + 1.65 = 18.15 \]
\[ 18.15 + 0.165 = 18.315 \]

### Final Answer:
The value of \( \sqrt{272.25} + \sqrt{2.7225} + \sqrt{0.027225} \) is \( 18.315 \).

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