Important Formulas: Indices and Surds | Quantitative Aptitude for SSC CGL PDF Download

Definition of Surds and Indices

• Surds: Numbers which can be expressed in the form √p + √q , where p and q are natural numbers and not perfect squares.
• Irrational numbers which contain the radical sign (n√) are called as surds Hence, the numbers in the form of √3, 3√2, ……. n√x in other words
• For example : √3, it can’t be simplified.
  √4 , it can be simplified so it is not a surds.
• Indices: Indices refers to the power to which a number is raised. For example; 3²

Types of Surds and Definitions

• Pure Surds: Those surds which do not have factors other than 1. For example 2√3, 3√7
• Mixed Surds: Those surds which do not have a factor of 1. For example √27 = 3√3, √50 = 5√2
• Similar Surds: When the radicands of two surds are the same. For example 5√2 and 7√2
• Unlike Surds: When the radicands are different. For example √2 and 2√5

Surds and Indices Rule

Surds and Indices Formulas

• (a + b)(a – b) = (a² – b²)
• (a + b)² = (a² + b² + 2ab)
• (a – b)² = (a² + b²- 2ab)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• (a³ + b³) = (a + b)(a² – ab + b²)
• (a³ – b³) = (a – b)(a²+ ab + b²)
• (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
• When a + b + c = 0, then a³ + b³ + c³ = 3abc.

Examples:
Q1: Solve the following expression using suitable algebraic identity: (2x + 3y)³

Solution:
To find: (2x + 3y)³
Using (a + b)³ Formula,
(a + b)³ = a³ + 3a²b + 3ab² + b³
(2x)³ + 3 × (2x)² × 3y + 3 × (2x) × (3y)² + (3y)³
8x³ + 36x²y + 54xy² + 27y³

Ans:
(2x + 3y)³ = 8x³ + 36x²y + 54xy² + 27y³

Q2: Ranbeer Kapoor wants to know the value of (256)^0.16 × (16)^0.18:

Solution:
Expression = (256)^0.16 × (16)^0.18
= (4)⁴ × 0.16 × (4)² × 0.18
= (4)^0.64 × (4)^0.36
= (4)¹
= 4