Is there a convention for power of a half being the positive square root?

Question

I know the $\surd$ sign refers to the positive square root. Does the exponent 1/2 mean the positive square root too by convention?

I ask because I'm converting from parametric to cartesian here...

$x=t^2$ and $y=t^3$

So $t=\pm \sqrt{x}$

Then $y=\pm x^{3/2}$

Yet the given answer is $y=x^{3/2}$ in the textbook. Can someone clarify please?

Thanks, Rob

Answer 1

A positive number to a real power is always, by convention, positive. This is because $a^b$ is generally defined as $e^{b \ln a}$.

In particular, in your case $t^{3/2}$ refers to $e^{(3/2) \ln t}$, a positive value.

It gets slightly more complicated if you have a negative number as a base, in which case for instance $a^{1/3}$ could mean the negative cube root, but $a^x$ is generally undefined for real $x$.