Not all entire radicals can be simplified and written as mixed radicals. The radicand must contain perfect nth root factors in order for it to be rewritten as a mixed radical.

For instance, «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»30«/mn»«/msqrt»«/math» is an entire radical. This entire radical is in its simplified form.

 

Example 4

Simplify «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»30«/mn»«/msqrt»«/math».

Step 1: Consider the factors of 30 (recall the Perfect Square chart you completed in the Check Up).

30 × 1, 15 × 2, 10 × 3, 6 × 5

Step 2: Identify the set of factors with the largest perfect square (other than 1).

For 30, there is no such set of factors. As such, the radical «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt mathcolor=¨#B94A48¨»«mn»30«/mn»«/msqrt»«/math» cannot be further simplified.

 

Is it possible to check if a number is a perfect square or a perfect cube?

Example 5

Is the number 36 a perfect square, a perfect cube, or neither?

Determine the prime factorization of the number 36.

Check which prime factors of 36 that can be listed in sets of squares (doubles) or cubes (triples).

36 = 2 × 2 × 3 × 3

36 = (2 × 2) × (3 × 3)

36 = 22 × 32

Since the sets are squares (doubles), and there are no single prime factors remaining, 36 is a perfect square.