Question

Use a factor tree to find the prime factors of 72. Write the prime factorization using exponents. ?

I confused about the exponents & prime factorization.

Answer 1

`2^3 xx 3^2`

Explanation:

`72 /2 = 2 xx 36`

`2 xx 36/2 = 2 xx 2 xx 18`

`2 xx 2 xx 18/2 = 2 xx 2 xx 2 xx 9`

`2 xx 2 xx 2 xx 9/3 = 2 xx 2 xx 2 xx 3 xx 3`

`2 xx 2 xx 2 xx 3 xx 3 = 2^3 xx 3^2`

an exponent gives the number of times a factor is multiplied by itself. 2 is multiplied by itself 3 times so `2^3`

Answer 2

`72 = 2xx2xx2xx3xx3 = 2^3 xx 3^2`

Explanation:

You want to write `72` as the product (multiplication) of the factors which are all prime numbers. (Numbers such as #2,3,5,7,11,...)

To use a factor tree ( also called the branch method), start with any two factors of `72` and then expand each until you get to a prime factor (they are shown in blue)

`color(white)(wwwwwwwwwwww)72`
`color(white)(wwwwxxxwwwww)"/"color(white)(x)"\"`
`color(white)(wwwwwwwwwww)8" "9`
`color(white)(wwwwx.xxx ww)"/ \"" ""/\"`
`color(white)(wwwwwwwww)4" "color(blue)(2" "3" "3)`
`color(white)(wwwwx.xww)"/ \"`
`color(white)(xxxxx.xxx)color(blue)(2" "2)`

From this we see `72 = 2xx2xx2xx3xx3 = 2^3 xx 3^2`

It does not matter which factors you start with.

`color(white)(wwwwwwwwwwww)72`
`color(white)(wwwwxxxwwwww)"/"color(white)(x)"\"`
`color(white)(wwwwwwwwwww)6" "12`
`color(white)(wwwwx.xxx ww)"/ \"" ""/\"`
`color(white)(wwwwwwwww)color(blue)(2" "3" "3)" "4`
`color(white)(wxxxxxxxx.wwwx.xww)"/ \"`
`color(white)(xxxxxxxxxxxxxx.xxx)color(blue)(2" "2)`

`72 = 2xx3xx3xx2xx2 = 2^3 xx 3^2`

Here is another possibility:

`color(white)(wwwwwwwwwwww)72`
`color(white)(wwwwxxxwwwww)"/"color(white)(x)"\"`
`color(white)(wwwwwwwwwww)color(blue)(2)" "36`
`color(white)(wwwwx.xxxwww. ww)"/ \"`
`color(white)(wwwwwwwwx.xxxw)6" "6`
`color(white)(wwwwx.xxxxx.ww)"/ \" " / \"`
`color(white)(xxxxxxxxxxxxx)color(blue)(2" "3color(white)(.)2" "3)`

`72 = 2xx2xx3xx2xx3 = 2^3 xx 3^2`