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Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

Writing a Prime Factorization with Powers

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 2 min 22 s
  • British Columbia
  • BC10-MA-04

A factor tree splits 360 into 36 × 10 and on down until every branch ends in a circled prime. The primes are collected as 2 × 2 × 2 × 3 × 3 × 5, checked as 8 × 9 × 5 = 360, and written with powers as 2³ × 3², with the base and exponent labelled, times 5. A second tree that starts from 8 × 45 gives the same primes, showing that each whole number has only one prime factorization, and the video ends on the common mistake of leaving the composite 9 in the answer.

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In one sentence

The question this lesson answers

Let us write 360 as a product of primes, called its prime factorization.

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

Transcript

What the lesson says

Let us write 360 as a product of primes, called its prime factorization.

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

A factor tree splits a number into two factors. 360 is 36 times 10.

36 is 6 times 6, and 10 is 2 times 5. 2 and 5 are prime, so circle them.

6 is composite: it has more than two factors. Split each 6 into 2 times 3.

Every branch now ends in a prime. Collect them: 2, 2, 2, 3, 3 and 5.

Check: the twos make 8, the threes make 9, and 8 times 9 times 5 is 360.

A power is short for a repeated factor: 2 times 2 times 2 is 2 to the power 3, or 2 cubed.

2 is the base. The exponent, 3, counts how many twos are multiplied.

3 times 3 is 3 to the power 2, or 3 squared. 5 is used once, so it stays 5.

So the prime factorization of 360 is 2 cubed times 3 squared times 5.

Now start with a different pair: 360 is also 8 times 45.

8 is 2 times 4, and 4 is 2 times 2. 45 is 9 times 5, and 9 is 3 times 3.

The primes are the same: three twos, two threes and one five.

Every whole number above 1 has only one prime factorization, apart from the order.

A common mistake is to stop at 9 and write 2 cubed times 9 times 5.

But 9 is composite: 9 is 3 times 3. Keep splitting until all are prime.

Recap: split into primes, then write repeats as powers: 2 cubed times 3 squared times 5.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Number: prime factorizationBC10-MA-04

Previous: Evaluating Powers with the Order of OperationsNext: Finding All the Factors of 72 with Factor Pairs

The outcomes and strands follow the British Columbia curriculum.

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