Simplifying Square Roots: √(√8100 + √6241) = ?

This problem asks us to find the value that replaces the question mark in the given mathematical expression. We need to simplify the expression using the properties of square roots.

Understanding the Expression

The expression is: $ \sqrt{\sqrt{8100} + \sqrt{6241}} $

Our goal is to evaluate this step-by-step.

Step 1: Calculate the inner square roots

First, let's find the square root of 8100:

• $ \sqrt{8100} = \sqrt{81 \times 100} = \sqrt{81} \times \sqrt{100} = 9 \times 10 = 90 $

Next, let's find the square root of 6241:

• To find $ \sqrt{6241} $, we can look for a number whose square is 6241. We know $ 70^2 = 4900 $ and $ 80^2 = 6400 $. Since 6241 ends in 1, the number must end in 1 or 9. Let's try 79.
• $ 79 \times 79 = (80 - 1) \times (80 - 1) = 80^2 - 2 \times 80 \times 1 + 1^2 = 6400 - 160 + 1 = 6241 $
• So, $ \sqrt{6241} = 79 $.

Step 2: Substitute the calculated values

Now, substitute these values back into the original expression:

• $ \sqrt{\sqrt{8100} + \sqrt{6241}} = \sqrt{90 + 79} $

Step 3: Perform the addition inside the square root

Add the numbers inside the remaining square root:

• $ 90 + 79 = 169 $

The expression becomes:

• $ \sqrt{169} $

Step 4: Calculate the final square root

Finally, calculate the square root of 169:

• We know that $ 13 \times 13 = 169 $.
• Therefore, $ \sqrt{169} = 13 $.

Conclusion

The value that comes in the place of the question mark ‘?’ is 13.

Evaluating the Options

Let's compare our result with the given options:

Our calculated value is 13, which matches option 4.