Explain why these expressions are equivalent: 0.5^(x-2) (0.5^(2) )/(0.5^(x) )

Question

Explain why these expressions are equivalent:


`0.5^(x-2)`



`(0.5^(2) )/(0.5^(x) )`

Answer 1

These are not equivalent.

If x=1, then

`0.5^(1-2) = 0.5^(-1) = 2`

but

`(0.5^2)/(0.5^1) = 0.5^1 = 0.5`

and those are not the same.

But if you were asking why
`0.5^(x-2) = (0.5^(x))/(0.5^(2))`, then the reason is this:

`\begin{aligned}0.5^(x-2) &= 0.5^(x) \cdot 0.5^(-2) \\[0.7em]&= 0.5^(x) \cdot (1)/(0.5^(2)) &\\[0.7em]&= (0.5^(x))/(0.5^(2))\end{aligned}`