Find the square root of 15625.

Introduction / Context:
This problem asks for the exact square root of 15625. Recognizing perfect squares and using factorization shortens the solution. Such questions are typical in number system and speed math sections and help improve mental calculation skills.

Given Data / Assumptions:
We are given the number 15625.We must find a positive number whose square equals 15625.We are looking for an exact integer square root, not an approximation.

Concept / Approach:
A good approach is to factor 15625 into prime factors or recognize it as a power of 5. Many powers of 5 end in 25, and 125 squared is a well known value. We can test 125 and confirm by direct multiplication, or factor systematically to be certain.

Step-by-Step Solution:
Step 1: Notice that 15625 ends with 625, which suggests a relationship with powers of 5.Step 2: Recall that 5^4 = 625 and 5^6 = 15625.Step 3: Therefore, 15625 = 5^6.Step 4: The square root of 5^6 is 5^(6 / 2) = 5^3.Step 5: Compute 5^3 = 5 × 5 × 5 = 125.Step 6: Thus, the square root of 15625 is 125.

Verification / Alternative check:
Verify by squaring 125 directly. Compute 125 × 125. First, 125 × 100 = 12500. Next, 125 × 25 = 125 × (100 / 4) = 12500 / 4 = 3125. Add these results: 12500 + 3125 = 15625. Since 125 squared equals 15625 exactly, our square root is correct.

Why Other Options Are Wrong:
If we square 135, we get 18225, which is larger than 15625. Squaring 145 gives 21025, also too large. Squaring 150 gives 22500, much larger than 15625. None of these match 15625, so they cannot be the square root.

Common Pitfalls:
Some learners guess nearby values and do not verify by squaring, which can lead to mistakes. Others may not recognize 15625 as a power of 5 and attempt lengthy factorization. Remembering common squares such as 125^2 = 15625 can speed up calculations significantly.

Final Answer:
The square root of 15625 is 125.