Here I will show you two methods that you can use to simplify the square root of 12356. In other words, I will show you how to find the square root of 12356 in its simplest radical form using two different methods.
To be more specific, I 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.
√12356 = A√B
Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 12356 to simplify the square root of 12356. 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 12356. The factors of 12356 are 1, 2, 4, 3089, 6178, and 12356. Furthermore, the greatest perfect square on this list is 4 and the square root of 4 is 2. Therefore, A equals 2.
B = Calculate 12356 divided by the greatest perfect square from the list of all factors of 12356. We determined above that the greatest perfect square from the list of all factors of 12356 is 4. Furthermore, 12356 divided by 4 is 3089, therefore B equals 3089.
Now we have A and B and can get our answer to 12356 in its simplest radical form as follows:
√12356 = A√B
√12356 = 2√3089
Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 12356 to simplify the square root of 12356 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 12356 and then take the square root of that product. The prime factors that multiply together to make 12356 are 2 x 2 x 3089. When we strip out the pairs only, we get 2 x 2 = 4 and the square root of 4 is 2. Therefore, A equals 2.
B = Divide 12356 by the number (A) squared. 2 squared is 4 and 12356 divided by 4 is 3089. Therefore, B equals 3089.
Once again we have A and B and can get our answer to 12356 in its simplest radical form as follows:
√12356 = A√B
√12356 = 2√3089