Simplify Square Root of 69012

Here we will show you two methods that you can use to simplify the square root of 69012. In other words, we will show you how to find the square root of 69012 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.

√69012 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 69012 to simplify the square root of 69012. 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 69012. The factors of 69012 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 71, 81, 108, 142, 162, 213, 243, 284, 324, 426, 486, 639, 852, 972, 1278, 1917, 2556, 3834, 5751, 7668, 11502, 17253, 23004, 34506, and 69012. Furthermore, the greatest perfect square on this list is 324 and the square root of 324 is 18. Therefore, A equals 18.

B = Calculate 69012 divided by the greatest perfect square from the list of all factors of 69012. We determined above that the greatest perfect square from the list of all factors of 69012 is 324. Furthermore, 69012 divided by 324 is 213, therefore B equals 213.

Now we have A and B and can get our answer to 69012 in its simplest radical form as follows:

√69012 = A√B

√69012 = 18√213

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 69012 to simplify the square root of 69012 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 69012 and then take the square root of that product. The prime factors that multiply together to make 69012 are 2 x 2 x 3 x 3 x 3 x 3 x 3 x 71. When we strip out the pairs only, we get 2 x 2 x 3 x 3 x 3 x 3 = 324 and the square root of 324 is 18. Therefore, A equals 18.

B = Divide 69012 by the number (A) squared. 18 squared is 324 and 69012 divided by 324 is 213. Therefore, B equals 213.

Once again we have A and B and can get our answer to 69012 in its simplest radical form as follows:

√69012 = A√B

√69012 = 18√213

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