INDEX FORM OF SURD

The index form of a surd n√a is

a1/n

For example, 35 can be written in index form as shown below.

35  = 51/3

What is surd ?

If ‘a’ is a positive rational number and n is a positive integer such that na is an irrational number, then nis called a ‘surd’ or a ‘radical’.

Order of a Surd

The general form of a surd is n√a is "√" is called a radical sign 'n' is called the order of the radical and 'a' is called radicand.

For example,

3√5 is a surd of order '3'

5√10 is a surd of order '5'

In the following table, the index form, order and radicand of some surds are given.

 Surd Index form Order Radicand √53√144√7√50 51/2141/371/4501/2 2342 514750

Practice Questions

Question 1 :

Convert the following surd to index form.

√7

Answer :

Surd  =  √7

Index form  =  71/2

Question 2 :

Convert the following surd to index form.

48

Answer :

Radical form  =  48

Index form  =  81/4

Question 3 :

Convert the following surd to index form.

36

Answer :

Radical form  =  36

Index form  =  61/3

Question 4 :

Convert the following surd to index form.

8

Answer :

Radical form  =  8√7

Index form  =  71/8

Laws of Surds

Law 1 :

n√a  =  a1/n

Law 2 :

n√(ab)  =  n√a x n√b

Law 3 :

n√(a/b)  =  n√a / n√b

Law 4 :

(n√a)n  =  a

Law 5 :

m√(n√a)  =  mna

Law 6 :

(n√a)m  =  n√am

Laws of Indices

Law 1 :

xm ⋅ xn  =  xm+n

Law 2 :

xm ÷ xn  =  xm-n

Law 3 :

(xm)n  =  xmn

Law 4 :

(xy)m  =  xm ⋅ ym

Law 5 :

(x / y)m  =  xm / ym

Law 6 :

x-m  =  1 / xm

Law 7 :

x0  =  1

Law 8 :

x1  =  x

Law 9 :

xm/n  =  y -----> x  =  yn/m

Law 10 :

(x / y)-m  =  (y / x)m

Law 11 :

ax  =  ay -----> x  =  y

Law 12 :

xa  =  ya -----> x  =  y

Laws of Surds and Indices - Practice Problems

Problem 1 :

Simplify the following :

√5  √18

Solution :

We have two radicals with same order. So, we can take the radical once and multiply the values inside the radicals.

√5  √18  =  √(5  18)

=  √(5  3 ⋅ 3 ⋅ 2)

=  3 (5  2)

=  3√10

Problem 2 :

Simplify the following :

3√35 ÷ 2√7

Solution :

We have two radicals with same order. So, we can take the radical once and divide the values inside the radicals.

3√35 ÷ 2√7  =  (3/2) ⋅ √(35/7)

=  (3/2) ⋅ √5

=  3√5/2

Problem 3 :

Simplify the following :

4√8 ÷ 4√12

Solution :

We have two radicals with same order. So, we can take the radical once and divide the values inside the radicals.

4√8 ÷ 4√12  =  4√(8/12)

=  4√(2/3)

=  4√2 / 4√3

Problem 4 :

Simplify the following :

x x3

Solution :

We have the same base 'x' for both the terms. Because the terms are multiplied, we can take the base once and add the exponents.

x x3  =  x2+3

x x3  =  x5

Problem 5 :

Simplify the following :

x7 ÷ x5

Solution :

We have the same base 'x' for both the terms. Because the terms are in division, we can take the base once and subtract the exponents.

x7 ÷ x5  =  x7-5

x7 ÷ x5  =  x2

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