Difference between Fractional Exponents and Fractions?

Question

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 $$4 * {{1/3}}$$

Answer 1

Depending on the level of student, the explanation might go as follows:

First, note that if $m$ and $n$ are integers, then we might regard exponentiation as repeated multiplication $$ x^m \cdot x^n = (\underbrace{x\cdot x\cdot \dotsb \cdot x}_{\text{$m$ times}})\cdot (\underbrace{x\cdot x\cdot \dotsb \cdot x}_{\text{$n$ times}}) = \underbrace{x\cdot x\cdot \dotsb \cdot x}_{\text{$m+n$ times}} = x^{m+n}. $$ This is a little bit of a lie (or not—again, it depends on the mathematical maturity of the audience). This can be worked out explicitly with small $m$ and $n$, and/or with fixed values of $x$ (say, work out $2^3\cdot 2^5$ by hand).

Then, when fractional exponents are introduced, it is reasonable to extend the operation above: if $p$ and $q$ are rational, then we should have $$ x^p\cdot x^q = x^{p+q}. $$ Again, with small $x$, $p$, and $q$, this could be justified by hand. Consider $$ 3^{\frac{1}{2}} \cdot 3^{\frac{5}{2}} = \sqrt{3}^1 \cdot \sqrt{3}^5 = \sqrt{3}^{5+1} = 3^{\frac{5+1}{2}} = 3^{\frac{5}{2} + \frac{1}{2}}. $$ This isn't really rigorous, but it should make the idea clear to students how have not seen it before. The moral of the story is that when terms with like bases are multiplied, the exponents are added.

This is in contrast to something like $$ \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} = \left( \frac{1}{3} \right)^4 = \frac{1}{3^4} = \frac{1}{81}. $$ Here we are again using the idea that exponentiation can be viewed as repeated multiplication. Compare this to something like $$ 3+3+3+3 = 4\cdot 3,$$ which is not the same as $$ 3+3+3+3 = 4+3. $$ Writing the second identity would be the same kind of error as writing $$ \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} = 4 \cdot \frac{1}{3}. $$

Answer 2

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 $$\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}=\frac{1\cdot 1\cdot 1\cdot 1}{3\cdot 3\cdot 3\cdot 3}=\frac{1}{81}.$$