Simplify Square Root of 75852

Here we will show you two methods that you can use to simplify the square root of 75852. In other words, we will show you how to find the square root of 75852 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.

√75852 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 75852 to simplify the square root of 75852. 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 75852. The factors of 75852 are 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 43, 49, 63, 84, 86, 98, 126, 129, 147, 172, 196, 252, 258, 294, 301, 387, 441, 516, 588, 602, 774, 882, 903, 1204, 1548, 1764, 1806, 2107, 2709, 3612, 4214, 5418, 6321, 8428, 10836, 12642, 18963, 25284, 37926, and 75852. Furthermore, the greatest perfect square on this list is 1764 and the square root of 1764 is 42. Therefore, A equals 42.

B = Calculate 75852 divided by the greatest perfect square from the list of all factors of 75852. We determined above that the greatest perfect square from the list of all factors of 75852 is 1764. Furthermore, 75852 divided by 1764 is 43, therefore B equals 43.

Now we have A and B and can get our answer to 75852 in its simplest radical form as follows:

√75852 = A√B

√75852 = 42√43

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 75852 to simplify the square root of 75852 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 75852 and then take the square root of that product. The prime factors that multiply together to make 75852 are 2 x 2 x 3 x 3 x 7 x 7 x 43. When we strip out the pairs only, we get 2 x 2 x 3 x 3 x 7 x 7 = 1764 and the square root of 1764 is 42. Therefore, A equals 42.

B = Divide 75852 by the number (A) squared. 42 squared is 1764 and 75852 divided by 1764 is 43. Therefore, B equals 43.

Once again we have A and B and can get our answer to 75852 in its simplest radical form as follows:

√75852 = A√B

√75852 = 42√43

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