Simplify Square Root of 82566

Here we will show you two methods that you can use to simplify the square root of 82566. In other words, we will show you how to find the square root of 82566 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.

√82566 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 82566 to simplify the square root of 82566. 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 82566. The factors of 82566 are 1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 99, 139, 198, 278, 297, 417, 594, 834, 1251, 1529, 2502, 3058, 3753, 4587, 7506, 9174, 13761, 27522, 41283, and 82566. Furthermore, the greatest perfect square on this list is 9 and the square root of 9 is 3. Therefore, A equals 3.

B = Calculate 82566 divided by the greatest perfect square from the list of all factors of 82566. We determined above that the greatest perfect square from the list of all factors of 82566 is 9. Furthermore, 82566 divided by 9 is 9174, therefore B equals 9174.

Now we have A and B and can get our answer to 82566 in its simplest radical form as follows:

√82566 = A√B

√82566 = 3√9174

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 82566 to simplify the square root of 82566 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 82566 and then take the square root of that product. The prime factors that multiply together to make 82566 are 2 x 3 x 3 x 3 x 11 x 139. When we strip out the pairs only, we get 3 x 3 = 9 and the square root of 9 is 3. Therefore, A equals 3.

B = Divide 82566 by the number (A) squared. 3 squared is 9 and 82566 divided by 9 is 9174. Therefore, B equals 9174.

Once again we have A and B and can get our answer to 82566 in its simplest radical form as follows:

√82566 = A√B

√82566 = 3√9174

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