Prime Factors, HCF and LCM

Recognising squares and cubes

• A number is a square number exactly when every index in its prime factor decomposition is even
  • 1225 = 5² × 7², so it is a square
• A number is a cube number exactly when every index is a multiple of 3
  • 3375 = 3³ × 5³, so it is a cube
• This test settles the question without any square rooting

Finding a square root from prime factors

• Halve every index, then multiply the result out
  • 1225 = 5² × 7², so its square root is 5 × 7 = 35
• The same idea works for a cube root by dividing every index by 3
  • 3375 = 3³ × 5³, so its cube root is 3 × 5 = 15

Finding an exact square root that is not an integer

• Where the indices are not all even, split off the largest even part of each
• Group the even-indexed primes together, since those come out of the root whole
• Whatever is left stays under the root sign as a surd
  • 300 = 2² × 3 × 5², which regroups as (2² × 5²) × 3
  • The square part gives 2 × 5 = 10, leaving √300 = 10√3
• This is how an exact answer is produced on the non-calculator papers, where a decimal would not be accepted as exact