Hi Michelle.

Any positive integer can be written as a unique product of prime numbers.

For example, to break 540 down to its prime factors, I simply start dividing by prime numbers:

• 540 is even, so 2 goes into it: 540 / 2 = 270.
• 270 is even, so 2 goes into it: 270 / 2 = 135.
• 3 goes into 135 (because 1+3+5=9 and 3 goes into 9), so 135 / 3 = 45.
• 5 goes into 45, so 45 / 5 = 9.
• 3 goes into 9, so 9 / 3 = 3.
• 3 goes into itself, so 3 / 3 = 1. When I get to 1, I stop.

Now I list what I divided by: 2, 2, 3, 5, 3, 3. When I multiply all these together I will get the original number:

2 × 2 × 3 × 5 × 3 × 3 = 540.

To write this in exponential form, I just re-order it to group the prime numbers together:

2 × 2 × 3 × 3 × 3 × 5

And then count the number of times each factor appears:

2² × 3³ × 5¹.

Usually, when an exponent is 1, we don't write it because it is assumed, so our final answer is:

2² × 3³ × 5.

Now you try it by breaking down 48 into its prime factors, Michelle.

Cheers,
Stephen La Rocque.