Simplify Square Root of 54978

Here we will show you two methods that you can use to simplify the square root of 54978. In other words, we will show you how to find the square root of 54978 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.

√54978 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 54978 to simplify the square root of 54978. 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 54978. The factors of 54978 are 1, 2, 3, 6, 7, 11, 14, 17, 21, 22, 33, 34, 42, 49, 51, 66, 77, 98, 102, 119, 147, 154, 187, 231, 238, 294, 357, 374, 462, 539, 561, 714, 833, 1078, 1122, 1309, 1617, 1666, 2499, 2618, 3234, 3927, 4998, 7854, 9163, 18326, 27489, and 54978. Furthermore, the greatest perfect square on this list is 49 and the square root of 49 is 7. Therefore, A equals 7.

B = Calculate 54978 divided by the greatest perfect square from the list of all factors of 54978. We determined above that the greatest perfect square from the list of all factors of 54978 is 49. Furthermore, 54978 divided by 49 is 1122, therefore B equals 1122.

Now we have A and B and can get our answer to 54978 in its simplest radical form as follows:

√54978 = A√B

√54978 = 7√1122

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 54978 to simplify the square root of 54978 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 54978 and then take the square root of that product. The prime factors that multiply together to make 54978 are 2 x 3 x 7 x 7 x 11 x 17. When we strip out the pairs only, we get 7 x 7 = 49 and the square root of 49 is 7. Therefore, A equals 7.

B = Divide 54978 by the number (A) squared. 7 squared is 49 and 54978 divided by 49 is 1122. Therefore, B equals 1122.

Once again we have A and B and can get our answer to 54978 in its simplest radical form as follows:

√54978 = A√B

√54978 = 7√1122

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