Eight Standard >> Exponents or power or index | Part -1

Product law of exponents

 We can write 
$2 \times 2 \times 2$=2^{3}\)
$2 \times 2 \times 2 \times 2$=2^{4}\)
$2 \times 2 \times 2 ......n times$=2^{n}\)

Rule 1. If \frac{p}{q} is any rational number then for any positive integer n we have $\left(\frac{p}{q}\right)^{n}=\frac{p^{n}}{q^{n}}$

Ex:  $\left(\frac{2}{3}\right)^{4}=\frac{2^{4}}{3^{4}}=\frac{16}{81}$
$\left(-\frac{4}{5}\right)^{3}=\frac{(-4) \times (-4) \times (-4)}{5 \times 5 \times 5}=\frac{-64}{125}$

Rule 2. For a rational number $\frac{p}{q}$, where $p \neq 0,\ q \neq 0,$ we have $\left(\frac{p}{q}\right)^{-1}=\frac{q}{p}$ [it is reciprocal or inverse of $\frac{p}{q}$]

 $\left(\frac{p}{q}\right)^{-n}=\left(\frac{q}{p}\right)^{n}$

Ex: $\left(-\frac{2}{3}\right)^{-3}=\left(-\frac{3}{2}\right)^{3}$
      $\left(\frac{2}{5}\right)^{-1}=\left(\frac{5}{2}\right)$