Solve \( 3w^{-\frac{2}{3}}+6=198\) for w.

I’m going to assume that you’ve done logarithms ("logs") in class
before you ran into this one on homework.

3(w to the -2/3 power) + 6= 198

Subtract 6 from each side:

3(w to the -2/3 power) = 192

Take the log of each side:

log(3) -(2/3) log(w) = log(192)

Subtract log(3) from each side:

-(2/3) log(w) = log(192) - log(3). notice that this side is log(192/3) = log(64)

-(2/3) log(w) = log(64)

Divide each side by -(2/3) :

log(w) = -(3/2) log(64)

Raise 10 to the power of each side:

w = (64) to the -3/2 power.

==> (A number) to the -3/2 power means 1/(the number to the +3/2 power).

==> The 3/2 power means either find the number’s square root and then cube it,
or else find the number’s cube and then square root it.

w =1 / (64 to the 3/2 power) = 1 / 512.


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