# CLASS-9FRACTIONAL INDICES

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