Answer:
9a^4 b^10(2 +3a^6 b^5) (2 -3a^6 b^5)
see work below
Step-by-step explanation:
36a^4b^10 - 81a^16b^20
A) find the GCF
36a^4b^10 = 4*9 a^4b^10 = 2*2*3*3 a*a*a*a*b*b*b*b*b*b*b*b*b*b
81a^16b^20= 9*9a^16b^20= 3*3*3*3* a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a *b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b*b
The terms that appear in both terms is the GCF. The terms that remain are inside the parentheses.
The GCF is 3*3 a*a*a*a*b*b*b*b*b*b*b*b*b*b
36a^4b^10 - 81a^16b^20 = 3*3 a*a*a*a*b*b*b*b*b*b*b*b*b*b *
( 2*2 - 3*3a*a*a*a*a*a*a*a*a*a*a*a b*b*b*b*b*b*b*b*b*b)
Combining like terms
36a^4b^10 - 81a^16b^20 = 9a^4b^10(4-9a^12b^10)
The expression inside the parenthesis can be factored using the difference of squares
let x^2 =4 x =2
y^2 = 9a^12 b^10 y = 3a^6b^5
(x^2 -y^2) = (x+y)(x-y)
9a^4b^10(4-9a^12b^10) = 9a^4b^10 ( 2+3a^6b^5) ( 2-3a^6b^5)
b) difference of squares a^2 – b^2 = (a + b)(a – b)
let a^2 = 36a^4b^10
so a = 6a^2b^5
b^2 = 81a^16b^20
b = 9a^8 b^10
a^2 – b^2 = (a + b)(a – b)
36a^4b^10 - 81a^16b^20 = (6a^2b^5 +9a^8 b^10) (6a^2b^5 -9a^8 b^10)
We can factor a 3 a^2 b^5 out of the first term
3 a^2 b^5 (2 +3a^6 b^5) (6a^2b^5 -9a^8 b^10)
3 a^2 b^5 (2 +3a^6 b^5) 3 a^2 b^5 (2 -3a^6 b^5)
Multiply the terms outside the parentheses together
9a^4 b^10(2 +3a^6 b^5) (2 -3a^6 b^5)
