Q:

Negative Fractional Indices

How do I figure this out:
$$\frac{27^{-2/3}}{16^{3/4}}$$
Please show workings. Thanks!

A:

Ok, so first we can see that both 27 and 16 can be written as powers of 3 and 2 respectively, so the expression in your question becomes $$\frac{(3^3)^{-\frac{2}{3}}}{(2^4)^{\frac{3}{4}}}$$ Next, we need to use a few index laws to simplify this. Remember that $(a^m)^n = a^{mn}$ and that $a^{-m} = \frac{1}{a^m}$. From this, we can simplify the above expression $$\frac{(3^3)^{-\frac{2}{3}}}{(2^4)^{\frac{3}{4}}} = \frac{3^{-2}}{2^{3}} = \frac{1}{3^22^3}=\frac{1}{72}$$

