Simplify Square Root of 19712

Here we will show you two methods that you can use to simplify the square root of 19712. In other words, we will show you how to find the square root of 19712 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.

√19712 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 19712 to simplify the square root of 19712. 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 19712. The factors of 19712 are 1, 2, 4, 7, 8, 11, 14, 16, 22, 28, 32, 44, 56, 64, 77, 88, 112, 128, 154, 176, 224, 256, 308, 352, 448, 616, 704, 896, 1232, 1408, 1792, 2464, 2816, 4928, 9856, and 19712. Furthermore, the greatest perfect square on this list is 256 and the square root of 256 is 16. Therefore, A equals 16.

B = Calculate 19712 divided by the greatest perfect square from the list of all factors of 19712. We determined above that the greatest perfect square from the list of all factors of 19712 is 256. Furthermore, 19712 divided by 256 is 77, therefore B equals 77.

Now we have A and B and can get our answer to 19712 in its simplest radical form as follows:

√19712 = A√B

√19712 = 16√77

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 19712 to simplify the square root of 19712 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 19712 and then take the square root of that product. The prime factors that multiply together to make 19712 are 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 7 x 11. When we strip out the pairs only, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 and the square root of 256 is 16. Therefore, A equals 16.

B = Divide 19712 by the number (A) squared. 16 squared is 256 and 19712 divided by 256 is 77. Therefore, B equals 77.

Once again we have A and B and can get our answer to 19712 in its simplest radical form as follows:

√19712 = A√B

√19712 = 16√77

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