Solving the Square Root Sum Equation

This problem requires us to find the sum of two square roots: $ \sqrt{29.8116} $ and $ \sqrt{77.44} $. We will calculate each square root separately and then add them together.

Step 1: Calculate the Square Root of 29.8116

We need to find the value of $ \sqrt{29.8116} $. Let's determine this value.

• We can estimate the value: $ 5^2 = 25 $ and $ 6^2 = 36 $. So, the square root is between 5 and 6.
• Testing values, we find that $ 5.46 \times 5.46 = 29.8116 $.

Therefore, $ \sqrt{29.8116} = 5.46 $.

Step 2: Calculate the Square Root of 77.44

Next, we need to find the value of $ \sqrt{77.44} $.

• We can estimate this value: $ 8^2 = 64 $ and $ 9^2 = 81 $. So, the square root is between 8 and 9.
• Testing values, we find that $ 8.8 \times 8.8 = 77.44 $.

Therefore, $ \sqrt{77.44} = 8.8 $.

Step 3: Sum the Calculated Square Roots

Now, we add the results from Step 1 and Step 2:

Sum = $ \sqrt{29.8116} + \sqrt{77.44} $

Sum = $ 5.46 + 8.8 $

Sum = $ 14.26 $

Step 4: Match the Result with the Options

The calculated sum is $ 14.26 $. Comparing this value with the given options:

• Option 1: 12.62
• Option 2: 12.26
• Option 3: 14.26
• Option 4: 14.62

Our calculated result $ 14.26 $ matches Option 3.