# How To Solve Surds And Indices Questions Quickly

## How to Solve Surds and Indices Questions Quickly

The natural number which cannot be expressed in the  form of fraction known as Surds. For example: $\sqrt{2}=2^{\frac{1}{2}}$ and the Indices refers to the power to which a number is raised.

### How to Solve Surds and Indices Ques. Quickly :

• Surds: Number which cannot be expressed in the fraction form of two integers is called as surd. For example: $\sqrt{2}$
• Indices: Indices refers to the power to which a number is raised. For example; 2²

## Sample Solutions Regarding the Rules of Surds and Indices :

Indices Multiplication rules:-

• Multiplication rule with same base
a n ⋅ a m = a(m+n)
Example:

23 ⋅ 24 = 2(3+4) = 27 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128

Surds Multiplication rules:-

• Multiplication rule with same indices
an ⋅ bn = (a ⋅ b)n
Example:
32 ⋅ 22 = (3⋅2)2 = 36

Indices Division rules:-

• Division rule with same indices
an / bn = (a / b)n
Example: 93 / 33 = (9/3)3 = 27

Surds Division rules:-

• Division rule with same base
am / an = a(m-n)
Example: 35 / 33 = 3(5-3) = 9

### Surds and Indices Power rules

Power rule 1

• (an)m = a(n.m)
Example:
(23)2 = 2(3.2) = 26 = 2⋅2⋅2⋅2⋅2⋅2 = 64

Power rule 2

• anm= a(nm) =
Example:
232 = 2(32) = 29 = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512

Power rule 3

• n√a =a(1/n)
Example:
3√27 = 27(1/3) = 3

### Type 1: How to Solve Surds and Indices Quickly- Simplify the expression

Question 1. Find the value of 7-25 – 7-26

Options:

A. 6 × 7-26

B. 6 × 7-25

C. 7 × 7-25

D. 7 × 7-26

Solution:    7-25 – 7-26 = $\frac{1}{7^{25}}$$\frac{1}{7^{26}}$

$\frac{7 – 1}{7^{26}}$ = 6 × 7-26

Correct option: A

Question 2. Simplify (256)$^ \frac{3}{4}$

Options:

A. 16

B. 12

C. 256

D. 64

Solution:    (256)$^ \frac{3}{4}$ = (44)$^ \frac{3}{4}$= 43 = 64

Correct option: D

Question 3. Find the value of 8112 ÷ 8110

Options:

A. 72

B. 64

C. 81

D. 49

Solution:    We know,

$\frac{a^m}{a^n}$ = am-n

= 8 (112 – 110) = 82 = 64

Correct option: B

### Type 2: Solve Surds and Indices Quickly- Find the value of x

Question 1. If 4x + 4x + 1 = 80, then the value of xx  is

Options:

A. 16

B. 9

C. 25

D. 4

Solution:     4x(1 + 4) = 80

4x  * 5 = 80

4x = $\frac{80}{5}$

4x = 16

x = 2

xx = 22 = 4

Correct option: D

Question 2. If 2a = 3$\sqrt{32}$, then a is equal to:

Options:

A. $\frac{1}{3}$

B. 4

C. $\frac{5}{3}$

D. $\frac{1}{2}$

Solution:    Given value 2a

3$\sqrt{32}$

2a = (32)$^ \frac{1}{3}$

2a = (25)$^ \frac{1}{3}$

2a = (2)$^ \frac{5}{3}$

a = $\frac{5}{3}$

Correct option: C

Question 3. If x and y are whole numbers such that xy = 169, then find the value of (x – 1)y + 1

Options:

A. 1331

B. 2744

C. 1728

D. 729

Solution:    169 = 132

So, the value of x = 13 and y = 2

Now, (x – 1)y + 1

= (13 – 1)2+1

= (12)

=1728

Correct option: C