Simplifying Square Roots and Cube Roots
Square Roots
We know:
4  2, since 22  4 and 9  3, since 32  9.
We also know that
8 won’t come out to a whole
number, but we can simplify it using 4 (because it
comes out to a whole number) in the following
way:
8  42
Cube Roots
We know:
3
8  2, since 23  8 and 3 27  3, since 33  27.
We also know that 3 16 won’t come out to a whole
number, but we can simplify it using 3 8 (because it
comes out to a whole number) in the following
way:
3
16  3 8  2
 4 2
 3 83 2
 2 2
We use this same principle (with other perfect
square numbers) in order to simplify other square
roots. Here are some other examples:
 23 2
We use this same principle (with other perfect cube
numbers) in order to simplify other cube roots.
Here are some other examples:
20  4  5
18  9  2
 4 5
 9 2
 3 83 4
 3 27 3 2
 2 5
 3 2
 23 4
 33 2
12  4  3
45  9  5
 4 3
 9 5
 3 83 5
 3 27 3 3
 2 3
 3 5
 23 5
 33 3
40  4 10
27  9  3
3
3
3
32  3 8  4
3
40  3 8  5
3
48  3 8  6
3
54  3 27  2
81  3 27  3
135  3 27  5
 4 10
 9 3
 3 83 6
 3 27 3 5
 2 10
 3 3
 23 6
 33 5
88  4  22
99  9 11
 4 22
 9 11
 3 8 3 11
 3 27 3 11
 2 22
 3 11
 2 3 11
 3 3 11
32  16  2
50  25  2
 16 2
 25 2
 3 125 3 2
 3 64 3 2
 4 2
 5 2
 53 3
 43 2
605  121 5
432  144  3
3
3
3
88  3 8 11
250  3 125  2
625  3 125  5
3
3
3
297  3 27 11
128  3 64  2
768  3 64 12
 121 5
 144 3
 3 125 3 5
 3 64 3 12
 11 5
 12 3
 53 5
 4 3 12
For radicals with an index higher than three, it can be helpful to write down a short list of integers raised to
the power of the index you are trying to simplify. For example when dealing with a fourth root:
24  16, 34  81, 44  256, 54 =625, . . .