Simplify Square Root of 78408

Here we will show you two methods that you can use to simplify the square root of 78408. In other words, we will show you how to find the square root of 78408 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.

√78408 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 78408 to simplify the square root of 78408. 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 78408. The factors of 78408 are 1, 2, 3, 4, 6, 8, 9, 11, 12, 18, 22, 24, 27, 33, 36, 44, 54, 66, 72, 81, 88, 99, 108, 121, 132, 162, 198, 216, 242, 264, 297, 324, 363, 396, 484, 594, 648, 726, 792, 891, 968, 1089, 1188, 1452, 1782, 2178, 2376, 2904, 3267, 3564, 4356, 6534, 7128, 8712, 9801, 13068, 19602, 26136, 39204, and 78408. Furthermore, the greatest perfect square on this list is 39204 and the square root of 39204 is 198. Therefore, A equals 198.

B = Calculate 78408 divided by the greatest perfect square from the list of all factors of 78408. We determined above that the greatest perfect square from the list of all factors of 78408 is 39204. Furthermore, 78408 divided by 39204 is 2, therefore B equals 2.

Now we have A and B and can get our answer to 78408 in its simplest radical form as follows:

√78408 = A√B

√78408 = 198√2

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 78408 to simplify the square root of 78408 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 78408 and then take the square root of that product. The prime factors that multiply together to make 78408 are 2 x 2 x 2 x 3 x 3 x 3 x 3 x 11 x 11. When we strip out the pairs only, we get 2 x 2 x 3 x 3 x 3 x 3 x 11 x 11 = 39204 and the square root of 39204 is 198. Therefore, A equals 198.

B = Divide 78408 by the number (A) squared. 198 squared is 39204 and 78408 divided by 39204 is 2. Therefore, B equals 2.

Once again we have A and B and can get our answer to 78408 in its simplest radical form as follows:

√78408 = A√B

√78408 = 198√2

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