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Lesson · Grade 8 · Mathematics

Spotting Perfect Squares and Cubes from Prime Factors

  • Grade 8
  • Mathematics
  • 2 min 13 s
  • British Columbia
  • BC8-MA-02

A factor tree breaks 144 into 2 × 2 × 2 × 2 × 3 × 3, with a warning not to stop at 4 because it is not prime. The primes split into two equal groups of 12, so 144 = 12². The same test shows that 216 splits into three equal groups of 6, so 216 = 6³, while 72 leaves one 2 over, so it is not a perfect square.

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In one sentence

The question this lesson answers

A perfect square is a whole number times itself: 5 times 5, or 5 squared, is 25.

A perfect cube is a whole number times itself times itself: 2 times 2 times 2, or 2 cubed, is 8.

Transcript

What the lesson says

A perfect square is a whole number times itself: 5 times 5, or 5 squared, is 25.

A perfect cube is a whole number times itself times itself: 2 times 2 times 2, or 2 cubed, is 8.

A prime number, like 2, 3, 5 or 7, has exactly two factors: 1 and itself.

Is 144 a perfect square? Start a factor tree: split it into 16 times 9.

9 is 3 times 3, and 3 is prime, so circle it.

16 is 4 times 4. But 4 is not prime, so do not stop here.

Each 4 is 2 times 2. Now every branch ends in a prime.

So 144 is 2 times 2 times 2 times 2 times 3 times 3.

Split the primes into two equal groups.

Each group is 2 times 2 times 3, which is 12.

So 144 is 12 times 12, or 12 squared: a perfect square.

Is 216 a perfect cube? Its prime factors are 2, 2, 2, 3, 3 and 3, since 8 times 27 is 216.

The primes split into three equal groups, each 2 times 3, which is 6.

So 216 is 6 times 6 times 6, or 6 cubed: a perfect cube.

Is 72 a perfect square? Its prime factors are 2, 2, 2, 3 and 3, since 8 times 9 is 72.

Three 2s cannot split equally into two groups, so one 2 is left over.

So 72 is not a perfect square.

Recap: prime factors in two equal groups mean a perfect square; in three equal groups, a perfect cube.

Curriculum

Where this fits

British Columbia, Grade 8

Mathematics, Number: perfect squares and cubesBC8-MA-02

Previous: Perfect squares and perfect cubes: building them from tiles and cubesNext: Square roots: the side of a square from its area

The outcomes and strands follow the British Columbia curriculum.

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