Simplify Square Root of 21546

Here we will show you two methods that you can use to simplify the square root of 21546. In other words, we will show you how to find the square root of 21546 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.

√21546 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 21546 to simplify the square root of 21546. 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 21546. The factors of 21546 are 1, 2, 3, 6, 7, 9, 14, 18, 19, 21, 27, 38, 42, 54, 57, 63, 81, 114, 126, 133, 162, 171, 189, 266, 342, 378, 399, 513, 567, 798, 1026, 1134, 1197, 1539, 2394, 3078, 3591, 7182, 10773, and 21546. Furthermore, the greatest perfect square on this list is 81 and the square root of 81 is 9. Therefore, A equals 9.

B = Calculate 21546 divided by the greatest perfect square from the list of all factors of 21546. We determined above that the greatest perfect square from the list of all factors of 21546 is 81. Furthermore, 21546 divided by 81 is 266, therefore B equals 266.

Now we have A and B and can get our answer to 21546 in its simplest radical form as follows:

√21546 = A√B

√21546 = 9√266

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 21546 to simplify the square root of 21546 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 21546 and then take the square root of that product. The prime factors that multiply together to make 21546 are 2 x 3 x 3 x 3 x 3 x 7 x 19. When we strip out the pairs only, we get 3 x 3 x 3 x 3 = 81 and the square root of 81 is 9. Therefore, A equals 9.

B = Divide 21546 by the number (A) squared. 9 squared is 81 and 21546 divided by 81 is 266. Therefore, B equals 266.

Once again we have A and B and can get our answer to 21546 in its simplest radical form as follows:

√21546 = A√B

√21546 = 9√266

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