How is y=1/2^x the same as y=2^-x

Answer 1

y=1/(2^x) = 2^-x

2^-x is the same as 1/(2^x)

In general, 1/a = a^-1. when you move a base from denominator to numerator the power changes sign

1/2^2 = 1/4 = 2^-2

a^b = 1/a^-b

to simplify a fraction with exponents, repeat the base and subtract the exponent in the denominator from the exponent in the numerator

a^0/a^2 = a^(0-2) = a^2 (a^0) = 1 as any base to the zero power = 1

1/(2^x) is the same as 2^0/2^x) = 2^(0-x) = 2^-x

use some specific values of x and you can see it work better

such as

1/2^3 = 1/8 = 2^0/2^3 = 2^(0-3) = 2^-3