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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 42 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, . . .