Question

How to show that `2^h=4/5`?

Given that `3(4^h)=4(2^k)` and `9(8^h)=20(4^k)`.

Answer 1

`4/5`

Explanation:

`3(4^h)=4(2^k)`

`3((2^2)^h) = 4(2^k)`

`3(2^{2h}) = 4(2^k)`

`3/4 = 2^{k-2h}`

`9(8^h)=20(4^k)`

`9(2^{3h}) = 20(2^{2k})`

`9/20 = 2^{2k - 3h}`

Here's the slightly tricky part. We need to make `h` out of a linear combination of the exponents. The easy combination that eliminates `k` works fine:

`h = (2k-3h)-2(k-2h)`

`2^h = {2^{2k-3h}}/(2^{k-2h})^2 = (9/20)/(3/4)^2 = 9/20 cdot 16/9 =4/5`