3
$\begingroup$

Why is the graph of $f(x) = (x^2)^\frac{1}{6}$ not the same graph as $f(x) = x^\frac{1}{3}$?

Shouldn't they be the same because when you apply the exponent rules to the first equation, you get the same result as the second equation?

By the way, I am an Algebra 2 student in High School.

Improve this question
$\endgroup$
  • 3
    $\begingroup$ Hint-When you square something the negative disappears. $\endgroup$ – KingLogic Nov 25 '20 at 2:43
  • 4
    $\begingroup$ Exponent rules don't work with negative bases and fractional exponents; cf. this question $\endgroup$ – J. W. Tanner Nov 25 '20 at 2:48
  • $\begingroup$ The short answer is that $(x^p)^q=x^{pq}$ only for $x>0$ when $p$ and $q$ are rational numbers. $\endgroup$ – Clayton Nov 25 '20 at 2:50
  • $\begingroup$ See also math.stackexchange.com/questions/2085268/does-i4-equal-1/…, and in particular the second-to-last paragraph. $\endgroup$ – mweiss Nov 25 '20 at 3:00
  • $\begingroup$ Genuine question: is $-1^ \frac13$ where the operation " ^ $^ \frac13$ " is acting upon real numbers only and not in the complex realm, generally considered to be equal to $-1$, or is it undefined? Or do different texts have different conventions? Ah, J. W. Tanner has linked to the answer to my question... $\endgroup$ – Adam Rubinson Nov 25 '20 at 3:07
5
$\begingroup$

When you square the quantity $x$, the negative sign disappears, and it doesn't come back if you raise it to the $\frac{1}{6}$ power. If you just take $\frac{1}{3}$ power, then the negative stays negative.

$((-1)^2)^\frac{1}{6}=1^\frac{1}{6}=1$, while $(-1)^\frac{1}{3}=-1$

Therefore, what you see on the first graph is the absolute value of the second graph.

This hyperlink contains a graph of the two equations. Note that when $x<0$ the red solid line is the blue dotted line reflected about the x-axis.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.