Question

How do you simplify `\frac { 3( x + 5) } { 5x ( x + 5) } \cdot \frac { 5} { 3( x - 3) }`?

Answer 1

`1/(x^2-3x)`

Explanation:

Since this is all multiplication, you can multiply the numerators and denominators. You will realize that there are a lot of similar terms which you can "cancel." These will be highlighted in various colours.

`(color(green)3color(red)((x+5)))/(color(blue)5xcolor(red)((x+5)))*color(blue)5/(color(green)3(x-3)`

To make this more clear, you can rearrange the expression so that the similar terms go on top of each other.

`(color(green)3color(red)((x+5)))/(color(green)3color(red)((x+5)))*color(blue)5/(color(blue)5x(x-3)`

Divisibility rules tell us that `x/x=1` if `x!=0`

Therefore, the term `(color(green)3color(red)((x+5)))/(color(green)3color(red)((x+5)))` is simply 1.

`=1*color(blue)5/(color(blue)5x(x-3)`

Note that we can also "cancel" or divide the two fives together. Remember that the division leaves the number `1` rather than it simply "disappearing."

`=cancelcolor(blue)5^1/(cancelcolor(blue)5^1x(x-3)`

`1/(1*x(x-3)`

Through the distributive property, `x(x-3)=x^2-3x`

Your final answer is `1/(x^2-3x)`