<s>[INST] Difference between Fractional Exponents and Fractions? how would you explain the difference between exponential multiplication and fractional multiplication?  $${x^{1/3}}{^{}{}} * {x^{1/3}}{^{}{}}  *{x^{1/3}}{^{}{}}  * {x^{1/3}}{^{}{}} = {x^{4/3}}$$ Why is this the same as $$4 * {^{1/3}}$$ On the other hand $$1/3 * 1/3 * 1/3 * 1/3 = 1/81$$ So in this case, this is not the same as $$4 * {{1/3}}$$
 [/INST] First, remember that $$b^x\cdot b^y=b^{x+y}.$$ That’s why $$x^{1/3}\cdot x^{1/3}\cdot x^{1/3}\cdot x^{1/3}=x^{1/3+1/3+1/3+1/3}=x^{4\cdot 1/3}=x^{4/3}.$$ One the other hand $$\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}=\frac{1\cdot 1\cdot 1\cdot 1}{3\cdot 3\cdot 3\cdot 3}=\frac{1}{81}.$$
</s>