Fractional Indices –
nth root of a : if ‘a’ is considered as any real number and ‘n’ is considered as a positive integer, then the nth root of ‘a’ is the real number ‘x’ such that xⁿ = a.
the nth root of a is denoted by a⅟ⁿ = ⁿ√a
Thus, ⁿ√a = x
=> a⅟ⁿ = x
=> a = xⁿ
There are some other symbol is given below –
i) √a = a⅟² is called the square root of a
ii) ᶟ√a = a⅓ is called the cube root of a
iii) ⁴√a = a⅟⁴ is called the 4th root of a
iv) ⁵√a = a⅟⁵ is called the 5th root of a
v) ⁶√a = a⅟⁶ is called the 6th root of a
please note – for positive value of a, the value of a⅟ⁿ will always be taken as positive.
there are some example are given below for your better understanding
Example.1) √9 = √3² = (3²)⅟² = 3
In above equation, 3 is base, 2 is index or exponent, and 2 is positive integers. Here, 2 X 1/2 = 1, so 3ˡ = 3
Example.2) ⁴√16 = ⁴√2⁴ = (2⁴)⅟⁴ = 2
In above equation, 2 is base, 4 is index or exponent, and 4 is positive integers. Here, 4 X 1/4 = 1, so 2ˡ = 2
Example.3) ⁵√27 = ⁵√3ᶟ = (3ᶟ)⅟⁵ = 3⅗
In above equation, 3 is base, 3 is index or exponent, and 5 is positive integers. Here, 3 X 1/5 = 3/5, so 3⅗
Example.4) ᶟ√64 = ᶟ√4ᶟ = (4ᶟ)⅓ = 4
Or, ᶟ√64 = ᶟ√2⁶ = (2⁶)⅓ = 2² = 2 X 2 = 4
In above equation, 4 is base, 6 is index or exponent, and 3 is positive integers. Here, 6 X 1/3 = 2, so 2²