Simplify Square Root of 19872

Here we will show you two methods that you can use to simplify the square root of 19872. In other words, we will show you how to find the square root of 19872 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.

√19872 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 19872 to simplify the square root of 19872. 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 19872. The factors of 19872 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 23, 24, 27, 32, 36, 46, 48, 54, 69, 72, 92, 96, 108, 138, 144, 184, 207, 216, 276, 288, 368, 414, 432, 552, 621, 736, 828, 864, 1104, 1242, 1656, 2208, 2484, 3312, 4968, 6624, 9936, and 19872. Furthermore, the greatest perfect square on this list is 144 and the square root of 144 is 12. Therefore, A equals 12.

B = Calculate 19872 divided by the greatest perfect square from the list of all factors of 19872. We determined above that the greatest perfect square from the list of all factors of 19872 is 144. Furthermore, 19872 divided by 144 is 138, therefore B equals 138.

Now we have A and B and can get our answer to 19872 in its simplest radical form as follows:

√19872 = A√B

√19872 = 12√138

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 19872 to simplify the square root of 19872 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 19872 and then take the square root of that product. The prime factors that multiply together to make 19872 are 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 23. When we strip out the pairs only, we get 2 x 2 x 2 x 2 x 3 x 3 = 144 and the square root of 144 is 12. Therefore, A equals 12.

B = Divide 19872 by the number (A) squared. 12 squared is 144 and 19872 divided by 144 is 138. Therefore, B equals 138.

Once again we have A and B and can get our answer to 19872 in its simplest radical form as follows:

√19872 = A√B

√19872 = 12√138

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