Factor the expression 36a^4(b^10) - 81a^16(b^20) using the two different techniques listed for Parts 1(a) and 1(b).
(a) Factor the given expression using the GCF monomial.
(b) Factor the given expression using the difference of squares
(a)
First, find the greatest common factor (GCF) of the terms:
GCF = 9a^4(b^10)
Now divide each term by the GCF:
36a^4(b^10) / 9a^4(b^10) = 4
81a^16(b^20) / 9a^4(b^10) = 9a^12(b^10)
So the factored expression using the GCF monomial is:
9a^4(b^10)(4 - 9a^12)
(b)
The given expression can be expressed as a difference of squares:
36a^4(b^10) - 81a^16(b^20) = (6a^2b^5)^2 - (9a^8b^10)^2
Now, using the formula for the difference of squares: a^2 - b^2 = (a + b)(a - b), the expression can be factored as:
(6a^2b^5 + 9a^8b^10)(6a^2b^5 - 9a^8b^10)