Basic Theory Part 2 for Indices and Surds for Upcoming Bank PO n SSC Exam

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’ = a³. 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 aⁿ. 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, 2³ = 2 x 2 x 2= 8 and 3⁴ = 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 (2³)², we first find out the value of the number inside the bracket (2³) as 8 and now raise this to the power 2. This gives 8² which is equal to 64. Thus (2³)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, 2⁴³ is evaluated by starting at the topmost level ‘3’. Thus we first calculate 4³ (which is 64). Since 2 is raised to the power 4³, we now have 2⁶⁴. .

Similarly, 2³² is equal to “2 raised to the power 3² “or “2 raised to the power 9” or 2⁹ 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 aⁿ = a^m	5² x 5⁷ = 5⁹
(2) = a^{m n}	= 7² = 49
(3) (a^m)ⁿ = a^mn	(4²)³ = 4⁶
(4) a^m =	2^-3 = = 0.125
(5)
(6) (ab)^m = a^m . b^m .	(2 x 3)⁴ = 2⁴ . 3⁴.
(7) a⁰ = 1	3⁰ = 1
(8) a¹ = a	4¹ = 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 2ⁿ = 2³, 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 a³= 4³ then a = 4.

(ii) if the powers are even numbers, then the bases are numerically equal but can have different signs. For example, if a⁴ = 3⁴ then a = + 3 or - 3. .