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.
Answer
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 $x \not=y$).
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