If `sqrt(4096)` = 64, then the value of `sqrt(40.96) + sqrt(0.4096) + sqrt(0.004096) + sqrt(0.00004096)` up to two places of decimals is :

Options

A. 7.09
B. `7.10`
C. 7.11
D. 7.12

Answer

To solve the problem, we need to calculate the following expression:

\[ \sqrt{40.96} + \sqrt{0.4096} + \sqrt{0.004096} + \sqrt{0.00004096} \]

We know that \(\sqrt{4096} = 64\). We can use this information to simplify the square roots of the decimal numbers.

### Step 1: Calculate \(\sqrt{40.96}\)

We can express \(40.96\) as:

\[ 40.96 = \frac{4096}{100} \]

Now, we can use the property of square roots:

\[ \sqrt{40.96} = \sqrt{\frac{4096}{100}} = \frac{\sqrt{4096}}{\sqrt{100}} = \frac{64}{10} = 6.4 \]

### Step 2: Calculate \(\sqrt{0.4096}\)

Next, we express \(0.4096\) as:

\[ 0.4096 = \frac{4096}{10000} \]

Using the same property of square roots:

\[ \sqrt{0.4096} = \sqrt{\frac{4096}{10000}} = \frac{\sqrt{4096}}{\sqrt{10000}} = \frac{64}{100} = 0.64 \]

### Step 3: Calculate \(\sqrt{0.004096}\)

Now, we express \(0.004096\) as:

\[ 0.004096 = \frac{4096}{1000000} \]

Again, using the property of square roots:

\[ \sqrt{0.004096} = \sqrt{\frac{4096}{1000000}} = \frac{\sqrt{4096}}{\sqrt{1000000}} = \frac{64}{1000} = 0.064 \]

### Step 4: Calculate \(\sqrt{0.00004096}\)

Finally, we express \(0.00004096\) as:

\[ 0.00004096 = \frac{4096}{100000000} \]

Using the square root property:

\[ \sqrt{0.00004096} = \sqrt{\frac{4096}{100000000}} = \frac{\sqrt{4096}}{\sqrt{100000000}} = \frac{64}{10000} = 0.0064 \]

### Step 5: Add all the results together

Now we can sum all the square roots we calculated:

\[ \sqrt{40.96} + \sqrt{0.4096} + \sqrt{0.004096} + \sqrt{0.00004096} = 6.4 + 0.64 + 0.064 + 0.0064 \]

Calculating this step by step:

1. \(6.4 + 0.64 = 7.04\)
2. \(7.04 + 0.064 = 7.104\)
3. \(7.104 + 0.0064 = 7.1104\)

### Final Result

Rounding \(7.1104\) to two decimal places gives us:

\[ \text{Final Answer} = 7.11 \]