Simplify Square Root of 65709

Here we will show you two methods that you can use to simplify the square root of 65709. In other words, we will show you how to find the square root of 65709 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.

√65709 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 65709 to simplify the square root of 65709. 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 65709. The factors of 65709 are 1, 3, 7, 9, 21, 49, 63, 147, 149, 441, 447, 1043, 1341, 3129, 7301, 9387, 21903, and 65709. Furthermore, the greatest perfect square on this list is 441 and the square root of 441 is 21. Therefore, A equals 21.

B = Calculate 65709 divided by the greatest perfect square from the list of all factors of 65709. We determined above that the greatest perfect square from the list of all factors of 65709 is 441. Furthermore, 65709 divided by 441 is 149, therefore B equals 149.

Now we have A and B and can get our answer to 65709 in its simplest radical form as follows:

√65709 = A√B

√65709 = 21√149

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 65709 to simplify the square root of 65709 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 65709 and then take the square root of that product. The prime factors that multiply together to make 65709 are 3 x 3 x 7 x 7 x 149. When we strip out the pairs only, we get 3 x 3 x 7 x 7 = 441 and the square root of 441 is 21. Therefore, A equals 21.

B = Divide 65709 by the number (A) squared. 21 squared is 441 and 65709 divided by 441 is 149. Therefore, B equals 149.

Once again we have A and B and can get our answer to 65709 in its simplest radical form as follows:

√65709 = A√B

√65709 = 21√149

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Simplify Square Root of 65710
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