Hello Mathematics Stack Exchange,
I'm currently a Grade $11$ Math Student and to train for this year's exam I am going through a worked video of the previous years' exams to get a better understanding of what to expect.
I came across the following question:
$\text{Consider the function}:$ $W=(x+2)^{\frac{2}{5}}$ Make $x$ the subject of the formula. Seems pretty straightforward, right?
Well, this is the process they went through to isolate $x$:
$W=(x+2)^{\frac{2}{5}}$
$x+2=W^{\frac{5}{2}}$
$x=W^{\frac{5}{2}}-2$
So from what I can see, to get rid of the exponent $\frac25$ they raised the other side of the equation to the reciprocal which is $\frac52$. Now, I have searched online for 'inverse of a fractional exponent' etc, and I haven't really come across anything. Could someone explain to me more clearly why they raised it to the power of '$\frac52$'. I mean I am happy with accepting it as it is but I don't really know why they did this.
If there's something wrong with the question please let me know.
Cheers, Tom
It comes from the exponent rule: $(x^a)^b = x^{ab}$. So if you want to "remove" an exponent, which means you're trying to get the final exponent to become $1$, you're trying to solve $ab = 1$ which means $b = \frac{1}{a}$. So if $a = \frac{2}{5}$, then $b = \frac{5}{2}$ meaning you have to raise everything to the power of $\frac{5}{2}$ to cancel out the power of $\frac{2}{5}$.