Simplify Square Root of 31824

Here we will show you two methods that you can use to simplify the square root of 31824. In other words, we will show you how to find the square root of 31824 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.

√31824 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 31824 to simplify the square root of 31824. 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 31824. The factors of 31824 are 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 17, 18, 24, 26, 34, 36, 39, 48, 51, 52, 68, 72, 78, 102, 104, 117, 136, 144, 153, 156, 204, 208, 221, 234, 272, 306, 312, 408, 442, 468, 612, 624, 663, 816, 884, 936, 1224, 1326, 1768, 1872, 1989, 2448, 2652, 3536, 3978, 5304, 7956, 10608, 15912, and 31824. Furthermore, the greatest perfect square on this list is 144 and the square root of 144 is 12. Therefore, A equals 12.

B = Calculate 31824 divided by the greatest perfect square from the list of all factors of 31824. We determined above that the greatest perfect square from the list of all factors of 31824 is 144. Furthermore, 31824 divided by 144 is 221, therefore B equals 221.

Now we have A and B and can get our answer to 31824 in its simplest radical form as follows:

√31824 = A√B

√31824 = 12√221

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 31824 to simplify the square root of 31824 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 31824 and then take the square root of that product. The prime factors that multiply together to make 31824 are 2 x 2 x 2 x 2 x 3 x 3 x 13 x 17. 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 31824 by the number (A) squared. 12 squared is 144 and 31824 divided by 144 is 221. Therefore, B equals 221.

Once again we have A and B and can get our answer to 31824 in its simplest radical form as follows:

√31824 = A√B

√31824 = 12√221

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