Aptitude Tricks – Surds and Indices

SURDS AND INDICES

 

Important Terms

Surd: Number which cannot be expressed in the fraction form of two integers is called as surd. Hence, the numbers in the form of 332, ……. nx

For example:  11/2 can be written as 1 but 3 cannot be written in the form of fraction
 91/2 3

Irrational numbers which contain the radical sign (     ) are called as surds.

Indices: Indices refers to the power to which a number is raised. Index is used to show that a number is repeatedly multiplied by itself.

For example: a3 is a number with an index of 3 and base ‘a’. It is called as “a to the power of 3”

Quick Tips and Tricks

1) The laws of indices and surds are to be remembered to solve problems on surds and indices. 

Laws of Indices

1) xm × xn = am+n

2) (xm)n = xmn

3) (xy) n = xnyn

4) xm = xm – n
xn
5) x n = xn
y yn
6) x–1 = 1
x

Laws of Surds

1) nx = x(1/n)

2) nxy = nx × ny

n(x/y) = nx
ny

4) (nx)n = X

5) mnx = mnx

6) (nx)m = (nxm)

2) Expressing a number in radical form

Example: l x(m/n) l = nxm

The exponential form l x(m/n) l is expressed in radical form as nxm

Important points to Remember 

1) Any number raised to the power zero is always equals to one. (Eg: x 0 = 1)
2) Surd nx can be simplified if factor of x is a perfect square
3) If denominator in a fraction has any surds, then rationalize the denominator by multiplying both numerator and denominator by a conjugate surd.
4) Every surd is an irrational number, but every irrational number is not a surd.
5) The conjugate of (2 + 7i) is (2 – 7i)
6) Different expressions can be simplified by rationalizing the denominator and eliminating the surd.

Rationalizing the denominator: 

To rationalize the denominator 7 multiply with its conjugate to both numerator and denominator

Example 1: 1 = 1 × 7 = 7
7 7 7 7

 

Example 2: 1 = 1 × 7 – 3 = 7 – 3
7 + 3 7 + 3 7 – 3 4

If 2^x × 8^(1/4) = 2^(1/4) then find the value of x ? (^=power)

If 9^x – (9)^(x – 1) = 648, then find the value of x^x

If 4^ (x – y) = 64 and 4 ^(x + y) = 1024, then find the value of x.

If a and b are whole numbers such that a^b = 121, then find the value of (a – 1)^(b + 1)

(32)^(n/5) × 2^(2n + 1)/ 4^n × 2^(n – 1)

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