Surds & Indices
Understanding Surds and Indices is essential for solving algebraic problems and simplifying expressions in competitive exams.
Surds
A surd is an irrational root of a number that cannot be simplified to remove the root. Common examples include √2, √3, √5, etc.
- √2, √3, √5 are surds.
- √4 = 2 is not a surd because it simplifies to a rational number.
- Surds cannot be expressed as exact fractions or decimals.
Properties of Surds
- √a × √b = √(a × b)
- √a ÷ √b = √(a ÷ b)
- (√a)² = a
- n√(a^m) = a^(m/n)
Indices (Exponents)
Indices or exponents represent repeated multiplication of a base number.
- a^m × a^n = a^(m+n)
- (a^m)^n = a^(m×n)
- a^0 = 1, provided a ≠ 0
- a^(-m) = 1 / a^m
- (ab)^m = a^m × b^m
Examples
- √8 = √(4 × 2) = 2√2
- (2^3)^4 = 2^(3×4) = 2^12
- a^3 × a^5 = a^(3+5) = a^8
- a^(-2) = 1 / a²
Previous Year Questions
- SSC 2019: Simplify √50.
5√2
- RRB JE 2020: Simplify (3^2)^4.
3^8 = 6561
- HSSC 2021: What is the value of a^0 for a ≠ 0?
1
- SSC CGL 2018: Simplify √18 + √50.
3√2 + 5√2 = 8√2
- RRB NTPC 2019: Simplify (2^4 × 2^5) ÷ 2^3.
2^(4+5-3) = 2^6 = 64
- SSC CHSL 2020: Find the value of (16)^(3/4).
(16)^(3/4) = (2^4)^(3/4) = 2^(4×3/4) = 2^3 = 8
- HSSC 2019: Simplify √(75) - √(27).
5√3 - 3√3 = 2√3
- SSC CGL 2021: Simplify (81)^(1/2) × (9)^(1/2).
9 × 3 = 27
- RRB JE 2018: Simplify a^(-3) × a^5.
a^(5-3) = a^2
- SSC MTS 2017: Simplify √32.
4√2
- HSSC JE 2020: Simplify (2^3)^0.
1
- SSC CGL 2020: Simplify (4^(1/2))^3.
(2)^3 = 8
- RRB NTPC 2021: Simplify (√3)^2.
3
- SSC CPO 2019: Simplify (9^(-1/2)).
1/3
- HSSC 2022: Simplify √(2) × √(8).
√16 = 4
- SSC MTS 2018: Simplify (25)^(3/2).
(5^2)^(3/2) = 5^3 = 125
- RRB JE 2022: Simplify a^4 ÷ a^6.
a^(4-6) = a^(-2) = 1 / a²
- SSC CGL 2017: Simplify √(98).
7√2
- HSSC JE 2017: Simplify (27)^(2/3).
(3^3)^(2/3) = 3^2 = 9
- SSC CHSL 2019: Simplify (a^5 × a^(-2))^3.
(a^3)^3 = a^9
Tip: Always simplify surds by factoring out perfect squares before performing operations.