Simplify Square Root of 77312

Here we will show you two methods that you can use to simplify the square root of 77312. In other words, we will show you how to find the square root of 77312 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.

√77312 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 77312 to simplify the square root of 77312. 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 77312. The factors of 77312 are 1, 2, 4, 8, 16, 32, 64, 128, 151, 256, 302, 512, 604, 1208, 2416, 4832, 9664, 19328, 38656, and 77312. Furthermore, the greatest perfect square on this list is 256 and the square root of 256 is 16. Therefore, A equals 16.

B = Calculate 77312 divided by the greatest perfect square from the list of all factors of 77312. We determined above that the greatest perfect square from the list of all factors of 77312 is 256. Furthermore, 77312 divided by 256 is 302, therefore B equals 302.

Now we have A and B and can get our answer to 77312 in its simplest radical form as follows:

√77312 = A√B

√77312 = 16√302

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 77312 to simplify the square root of 77312 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 77312 and then take the square root of that product. The prime factors that multiply together to make 77312 are 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 151. When we strip out the pairs only, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 and the square root of 256 is 16. Therefore, A equals 16.

B = Divide 77312 by the number (A) squared. 16 squared is 256 and 77312 divided by 256 is 302. Therefore, B equals 302.

Once again we have A and B and can get our answer to 77312 in its simplest radical form as follows:

√77312 = A√B

√77312 = 16√302

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