Simplify Square Root of 40176

Here we will show you two methods that you can use to simplify the square root of 40176. In other words, we will show you how to find the square root of 40176 in its simplest radical form using two different methods.

To be more specific, we have created an illustration below showing what we want to calculate. Our goal is to make "A" outside the radical (√) as large as possible, and "B" inside the radical (√) as small as possible.

√40176 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 40176 to simplify the square root of 40176. This is how to calculate A and B using this method:

A = Calculate the square root of the greatest perfect square from the list of all factors of 40176. The factors of 40176 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 31, 36, 48, 54, 62, 72, 81, 93, 108, 124, 144, 162, 186, 216, 248, 279, 324, 372, 432, 496, 558, 648, 744, 837, 1116, 1296, 1488, 1674, 2232, 2511, 3348, 4464, 5022, 6696, 10044, 13392, 20088, and 40176. Furthermore, the greatest perfect square on this list is 1296 and the square root of 1296 is 36. Therefore, A equals 36.

B = Calculate 40176 divided by the greatest perfect square from the list of all factors of 40176. We determined above that the greatest perfect square from the list of all factors of 40176 is 1296. Furthermore, 40176 divided by 1296 is 31, therefore B equals 31.

Now we have A and B and can get our answer to 40176 in its simplest radical form as follows:

√40176 = A√B

√40176 = 36√31

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 40176 to simplify the square root of 40176 to its simplest form possible. This is how to calculate A and B using this method:

A = Multiply all the double prime factors (pairs) of 40176 and then take the square root of that product. The prime factors that multiply together to make 40176 are 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 31. When we strip out the pairs only, we get 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 = 1296 and the square root of 1296 is 36. Therefore, A equals 36.

B = Divide 40176 by the number (A) squared. 36 squared is 1296 and 40176 divided by 1296 is 31. Therefore, B equals 31.

Once again we have A and B and can get our answer to 40176 in its simplest radical form as follows:

√40176 = A√B

√40176 = 36√31

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