Calculating the Square Root of 48: How-To & Steps

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  • 0:04 Steps to Solve
  • 2:10 Solution
  • 2:13 Application
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will show us how to find the square root of a number 'n' in general, then we will use the general process to find the square root of 48. To finish things up, we'll look at a real world application involving this process.

Steps to Solve

The number 48 is not a perfect square. When we want to find the square root of a number that isn't an obvious perfect square, we have a couple of options. Of course, we can always use a calculator and round to an acceptable decimal place. However, we don't always have a calculator handy, and the calculators on our cell phones may not have the square root function for us to use.

When this is the case, we can calculate √n by taking it through a few steps that will give us an exact answer. The answer will contain a square root if n isn't a perfect square, but it will simplify it more so its value can be more easily estimated. Let's take a look at these steps, and then use them to calculate the square root of 48.

In general, if we want to calculate √n, we follow these steps:

  1. Factor n as much as possible
  2. For any pairs of numbers in the factorization of n, move them outside of the square root as one copy of the number
  3. Simplify the result

Hmmm, if we look at these steps one by one, we see that each step really isn't too difficult, so if we just take them one at a time, finding √n is fairly simple! That's good news since we need to use these steps to find √48. Let's give it a try!

First, we want to factor 48 as much as possible. In other words, we want to find the prime factorization of 48. To do this, we break 48 into any two factors that multiply to 48. Then, we do the same for those factors and for the factors of those factors, and so on. We continue this process until we can't factor the factors anymore.


We see that 48 = 2⋅2⋅2⋅2⋅3, and step one is done! Moving on to step two, we want to pull pairs of numbers in the factorization out of the square root as one copy of the number. We see that we have two pairs of 2 in the factorization, so we pull out one copy of 2 for each of the pairs.


We get that √48 = 2⋅2√3. Lastly, step three says to simplify, so we go ahead and multiply the 2⋅2 outside of the square root to get 4√3. That's it! We get that √48 = 4√3.


The square root of 48 is 4√3.


Now, let's apply our new found knowledge on a practical level.

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