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.

**Radical Notation**

Let
*n* be a positive integer and *r* be a real number. If *r ^{n}
*=

* *^{n}*√x *=* r*

The
symbol * ^{n}√* (read as

**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 *r*^{2} = *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 = 2^{3}?)

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.

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

Consider
again results of the form *r* = * ^{n}*√

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}*√

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 2^{n }:

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

*Solution *

(i) 8 = 2 ×2 × 2 ; therefore 8 = 2^{3}

(ii) 32 = 2×2×2×2×2 = 2^{5}

(iii) 1/4 = 1/(2×2) = 1/2^{2} = 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/}*

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}*)

**Example 2.17 **

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

*Solution*

Tags : Real Numbers | Maths , 9th Maths : UNIT 2 : Real Numbers

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9th Maths : UNIT 2 : Real Numbers : Radical Notation | Real Numbers | Maths

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