A problem with exponent laws...

Solve for $x$:

$$x^{\frac 13}={32\over \sqrt{x}}$$

I'm not sure how start up this problem. I thought you had to multiply both sides by $\sqrt{x}$ so that it cancels out on the right side and moves to the other side and then convert it to a exponent to equal: $$x^{\frac 13} \times x^{\frac 12}=32$$ But due to exponent laws the two exponents are unable to add together because the denominators are different.

• Dragons answer is excellent. I think you are confusing denominator of the fraction with the bases of the powers as $a^x\times b^y\neq ab^{x+y}$ in general but $a^x \times a^y = a^{x+y}$
– Karl
Apr 1, 2015 at 22:03
• @dragon I wonder whether a completely new tag (exponents) is needed, when the tag (exponentiation) already exists. (If I did not miss anything, you have created the new tag.) Apr 2, 2015 at 10:15

You only need to recognize from exponent laws that $$x^{\frac 13}\times x^{\frac 12}=x^{\frac 13 + \frac 12}=x^{\frac 56}.$$
I think you are referring to the denominators of the fractions in the exponents. Yes, those denominators are different, but you can always find the common denominator first before adding the fractions: $$\frac 13 + \frac 12 = \frac 26 + \frac 36= \frac 56.$$
Finally, note that you can only add the exponents if you have the same base, as you do. For example, $x^{\frac 12}y^{\frac 13} \not= (xy)^{\frac 56}$ since $x$ and $y$ are different bases (assuming $x \not=y$).