Question

Simplify this problem

Answer 1

`(49 - (1)/(r^2) )/(7 - (1)/(r) )`

Rewrite the fraction as division:

`= (49 - (1)/(r^2)) / (7 - (1)/(r) )`

Make them into single fraction:

`= (49r^2 - 1)/(r^2) / (7r - 1)/(r)`

Change the divide fraction into multiplication fraction:

`= (49r^2 - 1)/(r^2) * (r)/(7r - 1)`

Factorise the difference of square a² - b² = (a + b) (a - b) :

`= ((7r+ 1)(7r - 1))/(r^2) * (r)/(7r - 1)`

Cancel the common factors:

`= ((7r+ 1))/(r)`

Answer 2

Comment. The best way to do this is to let 1/r = a and 1/r^2 = a^2. Now rewrite the problem.

Solution

`(49 - a^2)/(7 - a) \\ ((7 - a)(7 + a))/(7 - a)\text{ Notice a cancellation can take place}`

There is a 7 - a in both numerator and denominator, so that cancel providing a does not equal 7. a cannot equal 7 because that will put a 7 in the denominator and that makes the whole fraction = something over 0 which is undefined.

Answer
So far what we have is
7 + a

But a = 1/r
So the answer can be 7 + 1/r

r can be anything but 0 [for this answer]
and 1/7 for the cancellation.