If A = 3^ 9^ + 9^ 27^ + 27^ 81^ and B = 81^ - 3^ , then B A is equal to _____.

Step-by-step solution

Consider the general term x 3x . We can rewrite this by multiplying and dividing by 2 x : 2 x x 2 3x x = 2x 2 3x x Using 2x = (3x - x) = 3x x - 3x x , we get: 3x x - 3x x 2 3x x = 1 2 ( 3x 3x - x x ) = 1 2 ( 3x - x) Applying this identity to each term of A : 3^ 9^ = 1 2 ( 9^ - 3^ ) 9^ 27^ = 1 2 ( 27^ - 9^ ) 27^ 81^ = 1 2 ( 81^ - 27^ ) Adding these terms, the series telescopes: A = 1 2 ( 9^ - 3^ + 27^ - 9^ + 81^ - 27^ ) A = 1 2 ( 81^ - 3^ ) We are given B = 81^ - 3^ . Substituting B into the equation for A : A = 1 2