Question

Simplify:

`((a + 1/b)^x (a - 1/b)^y)/((b + 1/a)^x (b - 1/a)^y)`

Answer 1

We are asked to simplify:

`((a + 1/b)^x * (a - 1/b)^y)/((b + 1/a)^x * (b - 1/a)^y)`

Step 1: Rewrite denominator terms

`b + 1/a = (ab + 1)/a`,

`b - 1/a = (ab - 1)/a`

So denominator becomes:

`((ab + 1)/a)^x * ((ab - 1)/a)^y`

Step 2: Rewrite numerator terms

`a + 1/b = (ab + 1)/b`,

`a - 1/b = (ab - 1)/b`

So numerator becomes:

`((ab + 1)/b)^x * ((ab - 1)/b)^y`

Step 3: Form fraction

`(((ab + 1)/b)^x * ((ab - 1)/b)^y)/(((ab + 1)/a)^x * ((ab - 1)/a)^y)`

Step 4: Simplify

Separate terms:

= `((ab + 1)/b)^x/((ab + 1)/a)^x * ((ab - 1)/b)^y/((ab - 1)/a)^y`

Simplify each fraction:

• First fraction:
  `((ab + 1)/b)^x/((ab + 1)/a)^x = (a/b)^x`

• Second fraction:
  `((ab - 1)/b)^y/((ab - 1)/a)^y = (a/b)^y`

Step 5: Combine

`(a/b)^x * (a/b)^y = (a/b)^(x + y)`