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The given mathematical expression is:

\[
\sqrt{\frac{99^2 + 99}{99}}
\]

### Step-by-Step Solution:

1. **Simplify the expression inside the square root:**

   \[
   99^2 = 9801
   \]
   Now substitute this into the expression:

   \[
   \frac{99^2 + 99}{99} = \frac{9801 + 99}{99} = \frac{9900}{99}
   \]

2. **Simplify the fraction:**

   \[
   \frac{9900}{99} = 100
   \]

3. **Apply the square root:**

   \[
   \sqrt{100} = 10
   \]

### Final Answer:
\[
10
\]

Would you like further clarification on any part of this solution?

### Related Questions:
1. How do you simplify expressions with square roots?
2. What are some shortcuts for squaring numbers like 99?
3. How do you factor numbers quickly to simplify expressions like this one?
4. Can this expression be solved using algebraic identities?
5. How would you approach this problem if 99 were replaced by another number?

### Tip:
When simplifying square root expressions, always simplify the inside as much as possible before applying the square root.

Simplify the expression: sqrt((99^2 + 99) / 99)

Learn how to simplify the expression sqrt((99^2 + 99) / 99) using basic algebraic techniques, including squaring numbers and simplifying fractions. This step-by-step guide is ideal for students in Grades 8-10.

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