Eight Standard >> Exponents or power or index | Part -2

Laws of exponents or power or index on non-zero rational numbers

Law 1. \(a^{m}\times a^{n}=a^{m+n}\)
       \(3^{2}\times 3^{3}=3^{3+2}=3^{5}\)

Law 2. \(\frac{a^{m}}{a^{n}}=a^{m-n}\)
      \(\frac{3^{3}}{3^{2}}=3^{3-2}=3^{1}\)

Law 3.  \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\) (n>m)
      \(\frac{3^{3}}{3^{2}}=3^{3-2}=3^{1}\)

Law 4. \((a^{m})^{n}=a^{mn}\)
       \((2^{3})^{2}=2^{3 \times 2}=2^{6}=64\)

Law 5. i) \(a^{m}=\frac{1}{a^{m}}\)

 Ex:    \(2^{-3}=\frac{1}{2^{3}}\)

         ii) \(\frac{1}{a^{-m}}=a^{m}\)

 Ex:  \(\frac{1}{5^{-2}}=5^{-2}\)

Law 6. \(a^{0}=1\)
         
      \(a^{o}=a^{m-m}\)

Law 7. \((ab)^{m}=a^{m}.b^{n}\)
         \left(\frac{a}{b}\right)^{m}=\frac{a^{n}}{b^{n}}

Law 8. If \(x^{m}=x^{n}\)
\(\Rightarrow\ m=n\) [Provided \(x>0,\ x\neq 1\)]

Q. \(3^{2x}.3^{3}=1\). Find the value of x.

Solution:
       \(3^{2x}.3^{3}=1\)
   \(\Rightarrow\ 3^{2x+3}=3^{0}\)
    \(\Rightarrow\ 2x+3=0\)
    \(\Rightarrow\ 2x=-3\)
   \(\therefore\ x=-\frac{3}{2}\)