Radical Notation

Radical Notation

Let n be a positive integer and r be a real number. If rⁿ = x, then r is called the n^th root of x and we write

  ⁿ√x = r

The symbol ⁿ√ (read as nth root) is called a radical; n is the index of the radical (hitherto we named it as exponent); and x is called the radicand.

Note

It is worth spending some time on the concepts of the ‘square root’ and the ‘cube root’, for better understanding of surds.

What happens when n = 2? Then we get r² = x, so that r is ²√x, our good old friend, the square root of x. Thus ²√16 is written as √16 , and when n =3, we get the cube root of x, namely ³√x . For example, ³√8 is cube root of 8, giving 2. (Is not 8 = 2³?)

How many square roots are there for 4? Since (+2) × (+2) = 4 and also (–2)×(–2) = 4, we can say that both +2 and –2 are square roots of 4. But it is incorrect to write that √4 = ± 2 .

This is because, when n is even, it is an accepted convention to reserve the symbol ⁿ√x for the positive n^th root and to denote the negative n^th root by – ⁿ√x. Therefore we need to write √4 = 2 and −√4 = −2.

When n is odd, for any value of x, there is exactly one real n^th root. For example, ³√8 = 2 and ⁵√−32 = −2.

Thinking Corner

 Which one of the following is false?

(1) The square root of 9 is 3 or –3.

(2) √9 = 3

(3) −√9 = −3

(4) √9 = ±3

1. Fractional Index

Consider again results of the form r = ⁿ√x.  

In the adjacent notation, the index of the radical (namely n which is 3 here) tells you how many times the answer (that is 4) must be multiplied with itself to yield the radicand.

To express the powers and roots, there is one more way of representation. It involves the use of fractional indices.

We write ⁿ√x as x^1/n. 

With this notation, for example

 ³√64 is 64^1/3 and √25 is 25^1/2.

Observe in the following table just some representative patterns arising out of this new acquaintance:

Example 2.16

Express the following in the form 2ⁿ :

(i) 8 (ii) 32 (iii) 1/4 (iv) √2 (v) √8.

Solution

(i) 8 = 2 ×2 × 2 ; therefore 8 = 2³

(ii) 32 = 2×2×2×2×2 = 2⁵

(iii) 1/4 = 1/(2×2) = 1/2² = 2⁻²

(iv) √2 = 2^1/2

(v) √8 = √2×√2×√2 = (2^1/2)³ which may be written as 2^3/2

Meaning of x^{m/ n}, (where m and n are Positive Integers)

We interpret x^m/n either as the n^th root of the m^th power of x or as the m^th power of the n^th root of x.

In symbols, x^m/n = (x^m)^1/n or (x^1/n)^m = ⁿ√x^m or (ⁿ√x)^m

Example 2.17

Find the value of (i) 81^5/4 (ii) 64^-2/3

Solution