What is the value of `sqrt(729)+sqrt(72.9)+sqrt(7.29)` ?

Options

A. `40.5`
B. `45.6`
C. `33.5`
D. `38.23`

Answer

To solve the expression \( \sqrt{729} + \sqrt{72.9} + \sqrt{7.29} \), we will calculate each square root step by step.

### Step 1: Calculate \( \sqrt{729} \)
The number \( 729 \) can be expressed as \( 27^2 \) because:
\[ 27 \times 27 = 729 \]
Thus, we have:
\[ \sqrt{729} = 27 \]

**Hint for Step 1:** To find the square root of a perfect square, identify the number that when multiplied by itself gives the original number.

### Step 2: Calculate \( \sqrt{72.9} \)
To simplify \( \sqrt{72.9} \), we can express it as:
\[ \sqrt{72.9} = \sqrt{\frac{729}{10}} = \frac{\sqrt{729}}{\sqrt{10}} = \frac{27}{\sqrt{10}} \]

**Hint for Step 2:** When dealing with decimals, consider converting to a fraction to simplify the square root calculation.

### Step 3: Calculate \( \sqrt{7.29} \)
Similarly, we can express \( \sqrt{7.29} \) as:
\[ \sqrt{7.29} = \sqrt{\frac{729}{100}} = \frac{\sqrt{729}}{\sqrt{100}} = \frac{27}{10} = 2.7 \]

**Hint for Step 3:** Recognize that \( 100 \) is a perfect square, which simplifies the square root calculation.

### Step 4: Combine the results
Now we can combine all the results:
\[ \sqrt{729} + \sqrt{72.9} + \sqrt{7.29} = 27 + \frac{27}{\sqrt{10}} + 2.7 \]

### Step 5: Simplify the expression
To add these, we need a common denominator for the terms involving \( \sqrt{10} \):
\[ 27 + 2.7 + \frac{27}{\sqrt{10}} = 29.7 + \frac{27}{\sqrt{10}} \]

### Step 6: Approximate \( \sqrt{10} \)
Using \( \sqrt{10} \approx 3.16 \):
\[ \frac{27}{\sqrt{10}} \approx \frac{27}{3.16} \approx 8.54 \]

### Step 7: Final Calculation
Now add \( 29.7 + 8.54 \):
\[ 29.7 + 8.54 \approx 38.24 \]

Thus, the final value of \( \sqrt{729} + \sqrt{72.9} + \sqrt{7.29} \) is approximately \( 38.24 \).

### Conclusion
The closest option to our calculation is \( 38.23 \). Therefore, the answer is:
\[ \boxed{38.23} \]