If a number ‘a’ is taken three times and
added, then the sum is written as ‘three times a’ which is written as 3 x a =
3a. Instead of adding, if a is taken three times and multiply, the product is
written as ‘cube of a’ = a^{3}. We say that ‘a’ is expressed as an
exponent. Here, ‘a’ is called the ‘base’ and 3 is called the ‘power’ or ‘index’
or ‘exponent’. .
Similarly, ‘a’ can be expressed to any
exponent ‘n’ and accordingly written as a^{n}. This is read as “a to
the power n” or “a to the power of n” or “a raised to the power n.” .
For example, 2^{3} = 2 x 2 x 2= 8 and
3^{4} = 3 x 3 x 3 x 3 = 81. .
If a number raised to a certain power is
inside brackets and this is then raised to a power again, {i.e., a number of
the type (a^{m})^{n}  read as “a raised to the power m whole
raised to the power n”}, then the number inside the brackets is evaluated first
and then this number is raised to the power which is outside the brackets. .
For example, to evaluate (2^{3})^{2},
we first find out the value of the number inside the bracket (2^{3}) as
8 and now raise this to the power 2. This gives 8^{2} which is equal to
64. Thus (2^{3})^{2} is equal to 64. .
If we have powers in the manner of “steps”,
then such a number is evaluated by starting at the topmost of the “steps” and
coming down one “step” in each operation. .
For example, 2^{43} is evaluated by
starting at the topmost level ‘3’. Thus we first calculate 4^{3} (which
is 64). Since 2 is raised to the power 4^{3}, we now have 2^{64}.
.
Similarly, 2^{32} is equal to “2
raised to the power 3^{2} “or “2 raised to the power 9” or 2^{9}
which is equal to 512. .
There are certain basic rules/formulae for
dealing with numbers having powers. These are called Laws of Indices. The
important ones are listed below but we are not required to learn the proof for
any of these formulae/rules. .
Law of indices

Example

(1) a^{m} x a^{n} = a^{m}

5^{2} x 5^{7} = 5^{9}

(2) = a^{m n}

= 7^{2} = 49

(3) (a^{m})^{n} = a^{mn}

(4^{2})^{3} = 4^{6}

(4) a^{m} =

2^{3} = = 0.125

(5)


(6) (ab)^{m} = a^{m} . b^{m}
.

(2 x 3)^{4} = 2^{4} . 3^{4}.

(7) a^{0} = 1

3^{0} = 1

(8) a^{1} = a

4^{1} = 4

These rules/laws will help you in solving a
number of problems. In addition to the above, the student should also remember
the following rules: .
Rule 1:
When the bases of two EQUAL numbers are
equal, then their powers also will be equal. (If the bases are neither zero nor
± 1.) .
For example: If 2^{n} = 2^{3},
then it means n = 3
Rule 2:
When the powers of two equal numbers are
equal (and not equal to zero), two cases arise:
(i) if
the power is an odd number, then the bases are equal. For example, if a^{3 }=
4^{3} then a = 4.
(ii) if
the powers are even numbers, then the bases are numerically equal but can have
different signs. For example, if a^{4} = 3^{4} then a = + 3 or 
3. .