- #1

chwala

Gold Member

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- Homework Statement
- This is my own created problem;

Solve for ##x## given

##x^\frac{2}{3} - x^\frac{-3}{2}-6=0##

- Relevant Equations
- understanding of indices

The actual problem that i was looking at with my students was supposed to be

##x^\frac{2}{3} - x^\frac{-2}{3}-6=0##(which is easy to solve using quadratic equations) of which i wanted them to solve, ...then i realized then that i had erronously posted

##x^\frac{2}{3} - x^\frac{-3}{2}-6=0## on the board...and they were not able to proceed...

...anyway, i want to see if we can solve the problem as it is...##x^\frac{2}{3} - x^\frac{-3}{2}-6=0##

My take;

##x^\frac{2}{3}⋅x^\frac{3}{2}-1-6x^\frac{3}{2}=0##

##x^\frac{13}{6}-6x^\frac{3}{2}-1=0##

##x^\frac{2}{3} - x^\frac{-2}{3}-6=0##(which is easy to solve using quadratic equations) of which i wanted them to solve, ...then i realized then that i had erronously posted

##x^\frac{2}{3} - x^\frac{-3}{2}-6=0## on the board...and they were not able to proceed...

...anyway, i want to see if we can solve the problem as it is...##x^\frac{2}{3} - x^\frac{-3}{2}-6=0##

My take;

##x^\frac{2}{3}⋅x^\frac{3}{2}-1-6x^\frac{3}{2}=0##

##x^\frac{13}{6}-6x^\frac{3}{2}-1=0##

##\dfrac{13}{6}\log x-1.5 \log x= \log 6##

##\dfrac{2}{3}\log x=\log6##

##\dfrac{2}{3}\log x=0.778##

##\log x=1.167##

##x=10^{1.167}=14.689## (approximate solution in my opinion)

My calculator indicates the solution as ##x=14.7617##

Your input is welcome...