Evaluating the Square Root and Percentage Expression

The problem asks us to find the value of the expression \(\sqrt{21+\sqrt{49}+\sqrt{64}} \space {\%\:of\:5000}\). To solve this, we need to simplify the expression inside the square root first, then calculate the square root, and finally find the percentage of 5000.

Step-by-Step Calculation

Let's break down the calculation into simple steps:

1. First, we evaluate the innermost square roots: \(\sqrt{49}\) and \(\sqrt{64}\).
2. The square root of 49 is 7, because \(7 \times 7 = 49\). So, \(\sqrt{49} = 7\).
3. The square root of 64 is 8, because \(8 \times 8 = 64\). So, \(\sqrt{64} = 8\).
4. Now, substitute these values back into the expression inside the main square root:
5. The expression inside the main square root becomes \(21 + 7 + 8\).
6. Add these numbers together: \(21 + 7 + 8 = 36\).
7. So, the main expression simplifies to \(\sqrt{36} \space {\%\:of\:5000}\).
8. Next, evaluate the main square root: \(\sqrt{36}\).
9. The square root of 36 is 6, because \(6 \times 6 = 36\). So, \(\sqrt{36} = 6\).
10. Now the expression is \(6 \space {\%\:of\:5000}\).
11. This means we need to find 6 percent of 5000.
12. To calculate a percentage of a number, we can use the formula: \(\text{Percentage} \space {\%\:of\: \text{Number}} = \frac{\text{Percentage}}{100} \times \text{Number}\).
13. Using the formula, \(6 \space {\%\:of\:5000} = \frac{6}{100} \times 5000\).
14. Simplify the calculation: \(\frac{6}{100} \times 5000 = 6 \times \frac{5000}{100} = 6 \times 50\).
15. Finally, multiply 6 by 50: \(6 \times 50 = 300\).

Therefore, the value of the expression \(\sqrt{21+\sqrt{49}+\sqrt{64}} \space {\%\:of\:5000}\) is 300.