Mixed and Entire Radicals
­ A mixed radical is a radical that has a coefficient other than 1. This means that there is a number in front of the root sign. Ex. 4√3 or 22∛7.
­ An entire radical is a radical that has a coefficient of 1. This means that there isn't a number in front of the root sign. Ex. √17 or ∛14 or ∜15.
Changing from Entire to Mixed Radical
­ Like fractions, radicals can be simplified. With fractions, you use the greatest common factor to reduce both numerator and denominator. With radicals, we use perfect squares, perfect cubes, etc.
Ex. √24 (Since we are trying to square root a number that isn't a perfect square, in order to simplify it we must use a perfect square to reduce it. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The only perfect square in this list of factors is 4. Therefore, we break 24 down into the factors of 4 and 6.)
√24 = √4 x 6 = √4 x √6 = 2 x √6 = 2√6
Ex. ∛24 (Since we are trying to cube root a number that isn't a perfect cube, in order to simplify it we must use a perfect cube to reduce it. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The only perfect cube in this list of factors is 8. Therefore, we break 24 down into the factors of 8 and 3.)
∛24 = ∛8 x 3 = ∛8 x ∛3 = 2 x ∛3 = 2∛3
Changing from Mixed to Entire Radical
Ex. 4√3 (Since there is a square root sign, the coefficient 4 must have came from square rooting a number.)
4√3 = √16 x √3 = √16 x 3 = √48
Ex. 3∛2 (Since there is a cube root sign, the coefficient 3 must have came from cube rooting a number.)
3∛2 = ∛27 x ∛2 = ∛27 x 2 = ∛54 1
Simplifying Radicals ­ Strategy #2
If you have trouble simplifying radicals, then simplify using the following steps (more steps, but will work every time):
1. Break the number up into a factor tree (all prime factors)
2. Put all the prime factors under the radical sign.
3. For every pair of factors that are the same, take one out and cancel the other.
4. Multiply all the numbers that you took out and put that number before the radical sign.
5. Multiply all the numbers that you didn't take out or cancel and place that number under the radical sign.
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