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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.


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)3
=1728
Correct option: C
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