Base and Index:
If a real number is multiplied times in succession (where is a positive integer) then the product so obtained is called the -th power of and is written as (read as, to the power ). Thus, ….to factors. Here, is called the base of and is called the index or exponent of . For example, . Here, is the base and 4 is the exponent of .
Note: In particular, is called the square of and is called the cube of
If and are two real numbers and is a positive integer such that , then is called the -th root of and is written as or .
Hence, it can be clearly seen that the -th root of () is such a number which when multiplied multiplied by itself times, i.e. it is such a number whose -th power is equal to .
In particular, if , then is called the second root or square root of and is written as or or simply, .
If , then is called the third root or cube root of and it is written as or .
Square root of 25 is 5, i.e.
Cube root of 27 is 3, i.e.
Sixth root of 64 is 2, i.e.
So, it is very clear that 5 and (-5) are both square roots of 25. As a result, when we try to find the square root of a positive number , we actually mean . Similarly, when we try to find the cube root of a positive number , there are 3 roots of out of which only one is positive. In general, we have roots when we try to find the -th root of a positive number of which only one root is positive.
For simplicity, when we want to find the square root or cube root or -th root of a real positive number we shall always mean only the positive real root. So
2. If is a real negative number and:
(i) is an odd positive integer then there exists no psitive -th root of but we shall always get a real negative -th root of , say , such that . For example, if and , then or,
(ii) is an even positive integer then there exists no real number such that , i.e. in this case there is no real -th root of . For example has no real value, say , such that
Laws of Indices:
If a, b are two non-zero real numbers and m, n are positive integers then
This law is known as Fundamental Law of Index.
Since and are positive integers hence by definition we have,
Since and are positive integers, if $ m>n $, then is also a positive integer.
Hence by the positive law of index we have,
[ and are both positive integers]
Again, and are positive integers when . Hence, the law can be similarly proved when
By definition, we have,
[by fundamental law]