Simplify Square Root of 69972

Here we will show you two methods that you can use to simplify the square root of 69972. In other words, we will show you how to find the square root of 69972 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.

√69972 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 69972 to simplify the square root of 69972. 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 69972. The factors of 69972 are 1, 2, 3, 4, 6, 7, 12, 14, 17, 21, 28, 34, 42, 49, 51, 68, 84, 98, 102, 119, 147, 196, 204, 238, 294, 343, 357, 476, 588, 686, 714, 833, 1029, 1372, 1428, 1666, 2058, 2499, 3332, 4116, 4998, 5831, 9996, 11662, 17493, 23324, 34986, and 69972. Furthermore, the greatest perfect square on this list is 196 and the square root of 196 is 14. Therefore, A equals 14.

B = Calculate 69972 divided by the greatest perfect square from the list of all factors of 69972. We determined above that the greatest perfect square from the list of all factors of 69972 is 196. Furthermore, 69972 divided by 196 is 357, therefore B equals 357.

Now we have A and B and can get our answer to 69972 in its simplest radical form as follows:

√69972 = A√B

√69972 = 14√357

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 69972 to simplify the square root of 69972 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 69972 and then take the square root of that product. The prime factors that multiply together to make 69972 are 2 x 2 x 3 x 7 x 7 x 7 x 17. When we strip out the pairs only, we get 2 x 2 x 7 x 7 = 196 and the square root of 196 is 14. Therefore, A equals 14.

B = Divide 69972 by the number (A) squared. 14 squared is 196 and 69972 divided by 196 is 357. Therefore, B equals 357.

Once again we have A and B and can get our answer to 69972 in its simplest radical form as follows:

√69972 = A√B

√69972 = 14√357

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