Why can this fraction not be simplified further?

Question

I have the algebraic fraction:

a-1 ÷ 2a^3 + 6a^2

Apparently this is its most simplified form? But I thought, using index laws, I could subtract a^1 from the bottom a^3 (because their bases are the same), which would give me:

-1 ÷ 2a^2 + 6a^2 and I can then ADD the two denominator terms because they are LIKE:

-1 ÷ 8a^2 and therefore this would be the most simplified form?

Answer 1

That is not how to subtract fractions...

For example 1/2 - 1/1 is NOT (1-1)/(2-1) = 0/1 = 0

Instead you must find the common denominator: 1/2 - 1/1 = 1/2-2/2 = -1/2

THe denominator is 2a^3 - 6a^2 = (2a^2) ( a - 3)

Neither one of these factors cancels a-1