Question

Simplify the following:

`(2^(n + 2) xx 4^(n + 1))/(2^(n + 1) xx 4^(n - 2))`

Answer 1

Given: `(2^(n + 2) xx 4^(n + 1))/(2^(n + 1) xx 4^(n - 2))`

Step-wise calculation:

1. Express 4^{n + 1} and 4^{n – 2} in terms of base 2:

4 = 2²

⇒ 4^{n + 1} = (2²)^{n + 1}

⇒ 4^{n + 1} = 2^{2(n + 1)}

⇒ 4^{n + 1} = 2^{2n + 2}

4^{n – 2} = (2²)^{n – 2}

4^{n – 2} = 2^{2(n – 2)}

4^{n – 2} = 2^{2n – 4}

2. Substitute back into the expression:

`(2^(n + 2) xx 2^(2n + 2))/(2^(n + 1) xx 2^(2n - 4)) = (2^(n + 2 + 2n + 2))/(2^(n + 1 + 2n - 4))`

`(2^(n + 2) xx 2^(2n + 2))/(2^(n + 1) xx 2^(2n - 4)) = (2^(3n + 4))/(2^(3n - 3))`

3. Simplify the power of 2:

`2^(3n + 4) ÷ 2^(3n - 3) = 2^((3n + 4) - (3n - 3))`

`2^(3n + 4) ÷ 2^(3n - 3) = 2^7`

So, the simplified form is 128 or 2⁷.