LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

FRACTIONAL 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 4 ^{th} root of a**

**iv) ⁵√a = a⅟⁵ is
called the 5 ^{th} root of a**

**v) ⁶√a = a⅟⁶
is called the 6 ^{th} 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**