Question

(4^(n+1) + 6*4^n)^3/(8^(n+1) + 2*8^n)^2

Answer 1

Решение смотри в приложении

Answer 2

To simplify the expression, let's break it down step by step:

First, let's rewrite the expressions in terms of powers of 2:

4^(n+1) = (2^2)^(n+1) = 2^(2n+2) 4^n = (2^2)^n = 2^(2n) 64^n = 6(2^2)^n = 6*2^(2n)

8^(n+1) = (2^3)^(n+1) = 2^(3n+3) 8^n = (2^3)^n = 2^(3n) 28^n = 2(2^3)^n = 2*2^(3n)

Now, let's substitute these expressions back into the original equation:

(4^(n+1) + 64^n)^3 = (2^(2n+2) + 62^(2n))^3 (8^(n+1) + 28^n)^2 = (2^(3n+3) + 22^(3n))^2

Simplifying further:

(2^(2n+2) + 62^(2n))^3 = (2^(2n+2))^3 + 3(2^(2n+2))^2*(62^(2n)) + 3(2^(2n+2))(62^(2n))^2 + (62^(2n))^3 (2^(3n+3) + 22^(3n))^2 = (2^(3n+3))^2 + 2*(2^(3n+3))(22^(3n)) + (2*2^(3n))^2

Expanding and simplifying further:

(2^(2n+2))^3 = 2^(6n+6) (62^(2n))^3 = 2162^(6n) (2^(2n+2))^2 = 2^(4n+4) (62^(2n))^2 = 362^(4n) (62^(2n)) = 62^(2n)

(2^(3n+3))^2 = 2^(6n+6) (22^(3n))^2 = 42^(6n) (2^(3n+3))(22^(3n)) = 2^(3n+3+1)2^(3n+3) = 2^(6n+6) (22^(3n)) = 2*2^(3n)

Now, let's substitute these simplified expressions back into the original equation:

((2^(6n+6)) + (2162^(6n))) / ((2^(6n+6)) + (42^(6n))) = (2^(6n+6) * (1 + 216)) / (2^(6n+6) * (1 + 4)) = (2172^(6n+6)) / (52^(6n+6)) = (217/5) * 2^(6n+6 - (6n+6)) = (217/5) * 2^0 = 217/5

Therefore, the simplified expression is 217/5.