# Simplifying Square Roots: Finding the Value of √125

The square root of a number is the value that when multiplied by itself results in the original number. In other words, it is the number that can be squared to give the original number. In this case, what is the square root of 125?

To find the square root of 125, we must first understand how to calculate square roots of non-perfect squares, meaning numbers that are not perfect squares or that do not have integer square roots. In general, there are different methods to calculate square roots, such as long division or using a calculator, but one of the most common techniques is using prime factorization.

In prime factorization, we express the number we want to find the square root of as a product of its prime factors, and then group them by pairs to extract the square roots. For instance, the prime factors of 125 are 5 and 25, which are both perfect squares. Thus, we can write 125 as:

125 = 5 x 5 x 5
= 5^2 x 5

Then, we group the pairs of factors as follows:

125 = 5 x 5 x 5
= (5 x 5) x 5
= 25 x 5

Finally, we take the square root of the perfect square 25, which is equal to 5, and leave the non-square factor 5 outside:

√125 = √(25 x 5)
= √25 x √5
= 5√5

Therefore, the square root of 125 is 5√5, which is an irrational number, meaning that it cannot be expressed as a finite decimal or fraction.

In conclusion, finding the square root of non-perfect squares can be done using different methods, but prime factorization is often a helpful technique to simplify the process. In the case of 125, we can express it as 5 x 5 x 5, group the pairs of factors, and obtain 5√5 as the final result. Knowing how to calculate square roots can be useful not only for mathematical purposes but also for practical applications in areas such as engineering, physics, or computer science.