If \(a\) is a non-zero real number and \(m,\;n\) are real numbers, then

1. \(\left( {{a^m}} \right).\left( {{a^n}} \right) = {a^{m + n}}\)
   
2. \(\frac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\), in this result if we put \(m = n\), then we get \(\frac{{{a^m}}}{{{a^m}}} = {a^{m - m}}\)or \({a^0} = 1\). Thus any non-zero real number to the power 0 is 1.

3. \({\left( {{a^m}} \right)^n} = {a^{mn}}\), this can be proved by writing it in the following manner,

   \({a^m}.{a^m}.{a^m}........n\;{\rm{times}}\; = \;{a^{m + m + m + ...n\;{\rm{times}}}} = {a^{mn}}\)

   Note that \({({a^m})^n}\) and \({a^{{m^n}}}\) are not the same numbers, for example \({\left( {{2^5}} \right)^4} = {2^{20}}\), but \({2^{{5^4}}} \ne {2^{20}}\). 

4. \({a^{ - m}} = \frac{1}{{{a^m}}}\), for example \({2^{ - 3}} = \frac{1}{{{2^3}}} = \frac{1}{8}\)
   
5. \({a^{m/n}}\)is \({n^{th}}\) root of \({a^m}\), \(m\) is any integer and \(n\) is any positive integer.
   

6. The result  \({a^m}{b^m} = {(ab)^m}\), is not always true. For example, 

   \(m = \frac{1}{2},\;a =  - 2,\;b =  - 3\)

   L.H.S. = \(\sqrt { - 2} \sqrt { - 3}  = \sqrt {2i} \sqrt {3i} \)

   or \(\sqrt 6  = \sqrt 6 {i^2}\) or \(\sqrt 6  =  - \sqrt 6 \), 

   That is a contradiction. Thus this rule is not valid for imaginary numbers

7. For the solution of \({a^x} = {a^y}\) where \(a,x,y\) are real numbers, consider the following cases carefully.

   (a) If \(a = 1\), then \(x,y\) may be any real numbers,

   (b) If \(a =  - 1\), then \(x,y\) may be both even or both odd.

   (c) If \(a = 0\), then \(x,y\) may be any non–zero real numbers.

   (d) If \(a\) is not equal to 0, 1 or – 1, then \(x = y\)

8. For the solution of \({a^x} = {b^x}\), \(a,b,x\) are real numbers. Consider the following cases carefully:

   (a) If \(a \ne  \pm b\),  then \(x = 0\)

   (b) If \(a = b \ne 0\), then \(x\) may have any real value

   (c) If \(a =  - b\), then \(x\) is even.

Example 01: Simplify the expression

\({\left( {\frac{{{x^a}}}{{{x^b}}}} \right)^{{a^2} + ab + {b^2}}}{\left( {\frac{{{x^b}}}{{{x^c}}}} \right)^{{b^2} + bc + {c^2}}}{\left( {\frac{{{x^c}}}{{{x^a}}}} \right)^{{c^2} + ca + {a^2}}}\)

Example 02: Solve the equation \({4^{x + 2}} = {2^{2x + 1}} + 14\), where \(x\) is a real number.

Example 03: If \({2^{{2^{10}}}} + {2^{{2^{10}}}} = {2^m}\), then the value of \(m\) is: