Simplify Square Root of 78078

Here we will show you two methods that you can use to simplify the square root of 78078. In other words, we will show you how to find the square root of 78078 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.

√78078 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 78078 to simplify the square root of 78078. 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 78078. The factors of 78078 are 1, 2, 3, 6, 7, 11, 13, 14, 21, 22, 26, 33, 39, 42, 66, 77, 78, 91, 143, 154, 169, 182, 231, 273, 286, 338, 429, 462, 507, 546, 858, 1001, 1014, 1183, 1859, 2002, 2366, 3003, 3549, 3718, 5577, 6006, 7098, 11154, 13013, 26026, 39039, and 78078. Furthermore, the greatest perfect square on this list is 169 and the square root of 169 is 13. Therefore, A equals 13.

B = Calculate 78078 divided by the greatest perfect square from the list of all factors of 78078. We determined above that the greatest perfect square from the list of all factors of 78078 is 169. Furthermore, 78078 divided by 169 is 462, therefore B equals 462.

Now we have A and B and can get our answer to 78078 in its simplest radical form as follows:

√78078 = A√B

√78078 = 13√462

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 78078 to simplify the square root of 78078 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 78078 and then take the square root of that product. The prime factors that multiply together to make 78078 are 2 x 3 x 7 x 11 x 13 x 13. When we strip out the pairs only, we get 13 x 13 = 169 and the square root of 169 is 13. Therefore, A equals 13.

B = Divide 78078 by the number (A) squared. 13 squared is 169 and 78078 divided by 169 is 462. Therefore, B equals 462.

Once again we have A and B and can get our answer to 78078 in its simplest radical form as follows:

√78078 = A√B

√78078 = 13√462

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