Simplify Square Root of 31500

Here we will show you two methods that you can use to simplify the square root of 31500. In other words, we will show you how to find the square root of 31500 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.

√31500 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 31500 to simplify the square root of 31500. 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 31500. The factors of 31500 are 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 25, 28, 30, 35, 36, 42, 45, 50, 60, 63, 70, 75, 84, 90, 100, 105, 125, 126, 140, 150, 175, 180, 210, 225, 250, 252, 300, 315, 350, 375, 420, 450, 500, 525, 630, 700, 750, 875, 900, 1050, 1125, 1260, 1500, 1575, 1750, 2100, 2250, 2625, 3150, 3500, 4500, 5250, 6300, 7875, 10500, 15750, and 31500. Furthermore, the greatest perfect square on this list is 900 and the square root of 900 is 30. Therefore, A equals 30.

B = Calculate 31500 divided by the greatest perfect square from the list of all factors of 31500. We determined above that the greatest perfect square from the list of all factors of 31500 is 900. Furthermore, 31500 divided by 900 is 35, therefore B equals 35.

Now we have A and B and can get our answer to 31500 in its simplest radical form as follows:

√31500 = A√B

√31500 = 30√35

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 31500 to simplify the square root of 31500 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 31500 and then take the square root of that product. The prime factors that multiply together to make 31500 are 2 x 2 x 3 x 3 x 5 x 5 x 5 x 7. When we strip out the pairs only, we get 2 x 2 x 3 x 3 x 5 x 5 = 900 and the square root of 900 is 30. Therefore, A equals 30.

B = Divide 31500 by the number (A) squared. 30 squared is 900 and 31500 divided by 900 is 35. Therefore, B equals 35.

Once again we have A and B and can get our answer to 31500 in its simplest radical form as follows:

√31500 = A√B

√31500 = 30√35

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