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 Question from Emma, a student: I need to find out how to make a prime factorization of 120 in exponential form.

Hi Emma,

I want to illustrate the process with a different number, 1575.

The smallest prime is 2 so try to divide 1575 by 2. 2 doesn't divide 1575 so try the next prime, 3. $\frac{1575}{3} = 525$ so $1575 = 3 \times 525$. Now divide 525 by 3, $\frac{525}{3} = 175$ so $1575 = 3 \times 3 \times 175.$ 175 isn't divisible by 3 so try the next prime, 5. $\frac{175}{5} = 35$ so $1575 = 3 \times 3 \times 5 \times 35.$ $35 = 5 \times 7$ so I have the prime factorization of 1575.

$1575 = 3 \times 3 \times 5 \times 5 \times7.$

Written in exponential form that's

$1575 = 3^2 \times 5^2 \times 7.$

I could have written $7$ as $7^1$ but usually when the exponent is 1 we don't write it.

Now you try 120,
Penny

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