can anyone explain why 2 to the neg 3 power is 1/2 to the 3 power?

because there can't be a negative exponent(-3 power) in the numerator of a number. You have to bring it down into the denominator(1/2 to the third)

2^-3
= 1/2^3 [= (1/2)^3]
= 1/8

P.S.
a^-n
= 1/a^n

Basically, the idea is For exponents, you want it to follow ordinary rules, such as

2^2 * 2^3 = 2^(2+3)=2^5

because that is how multiplcation works. In order for this rule to work in regards to negative exponents, negative exponents are given the effect of doing repeated division to invert the normal process of repeated multiplication. So Then,

2^4 * 2^(-2) = 2^2 = 4

Which is equivilents to thinking about it as

2^4/2^2 = 2^2 = 4

For the same reason, an exponent to the 0 power is 1.
2^2 * 2^(-2) = 2^0 = 2^2/2^2 = 1

laws of indices

a^-n = 1/(a^n)

a negative power means its one over constant^n

if a number is ever to the negative power, all you do is take the number to the other side of the denoninator and change the negative power to a postive one.

in this case you have (2/1)^-3. change it to (1/2^3). this is an identity. they are the same. this is the same as (1/2)^3 also.

hope this helps :)

to change negative power to positive power u invert the base.
any power changes number k to 1/k or vice versa
e.g.
2^(-3) = 1/2 ^3
5^(-4) = 1/5 ^4

neg power indicates it is the inverse and hence the reciprocal.

Andrew, revise your indices man. You forgot a^(-n) = 1/(a^n)

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

A negative exponent reciprocates the base and becomes positive.