Final answer:
The difference between finding the square root and cube root of a number using a factor tree lies in the grouping of prime factors: pairs for square roots and trios for cube roots. Square roots require pairs of factors because squares are products of a number by itself, while cube roots require groupings of three because cubes are products of a number multiplied by itself twice.
Step-by-step explanation:
The difference in procedure for finding the square root of a number using a factor tree and finding the cube root of a number using a factor tree lies in the grouping of the prime factors derived from the factor tree. When finding the square root, after breaking the number into its prime factors, you group the factors in pairs, since the square root of a number is a number which, when multiplied by itself, gives the original number. That is, √(x²) = x. Each pair of same prime factors represents one factor of the square root.
For the cube root, you group factors in threes because the cube root of a number is a number which, when cubed, gives the original number. That is, ∛(x³) = x. Each trio of the same prime factors corresponds to one factor of the cube root. If there are not enough factors to make a complete group of two (for square roots) or three (for cube roots), then the number inside the root symbol is not a perfect square or perfect cube, and a decimal or irrational number is expected.
For instance, the prime factorization of 32 is 2 x 2 x 2 x 2 x 2. To find its square root, you would pair the twos to get (2 x 2) x (2 x 2), and since we have an extra 2, the square root of 32 is not a whole number. However, when finding the cube root, you group them as (2 x 2 x 2) and an extra 2 x 2, meaning the cube root of 32 is 2 with a remainder of 2 x 2, or as a decimal, approximately 3.2.