How do you find the exact square root of 841?

To find the exact square root of 841, follow these detailed steps:

Understand the problem: The goal is to determine the number that, when multiplied by itself (squared), equals 841.

Check for perfect squares: Start by recognizing whether 841 is a perfect square. A perfect square is a number that has an integer as its square root.

Use prime factorization:

Begin by finding the prime factors of 841.
Check if 841 is divisible by smaller prime numbers like 2, 3, 5, etc. Since 841 is odd, it is not divisible by 2.
Check divisibility by 3 and 5, which it also fails.
Try dividing by 7, 11, and other successive primes until reaching 29:
841 ÷ 29 = 29, confirming 29 × 29 = 841.
Therefore, 841 =
2
9
2
29
2
.
Confirm the result:

Multiply 29 by itself to ensure accuracy: 29 × 29 = 841.
Conclusion:

The square root of 841 is 29.
This is the exact value, as 841 is a perfect square with 29 as its square root.