Question

How do you multiply and divide `\frac { x ^ { 2} - 3x } { x ^ { 2} + 3x - 10} \cdot \frac { 2x + 10} { 3} \div \frac { x ^ { 2} - x - 6} { x ^ { 2} - 4}`?

Answer 1

`2/3x`

Explanation:

`color(blue)("Preliminary thoughts")`

The trick is to 'play' with the formats looking for things to cancel out.

Note that:`" "x^2-4" is the same as "x^2-2^2=(x-2)(x+2)`
Also the shortcut for divide is turn upside down (invert) and multiply.

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`color(blue)("Simplifying the expression")`

Given`" "(x^2-3x)/(x^2+3x-10)color(white)(.)xx(2x+10)/3-:(x^2-x-6)/(x^2-4)`

Write as `(x(x-3))/((x-2)(x-5)) xx(2(x+5))/3xx((x-2)(x+2))/((x-3)(x+2))`

`color(white)(.)`

Cancelling out `(x cancel((x-3)))/(cancel((x-2))cancel((x-5))) xx(2cancel((x+5)))/3xx(cancel((x-2))cancel((x+2)))/(cancel((x-3))cancel((x+2)))`

So this simplifies to `" "2/3x`