Definition and Types of Surds and Indices
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 (√) are called surds. For example: √3, 3√2, etc.
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 have a factor other than 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
| Rule Name | Surds Rule | Indices Rule |
|---|---|---|
| Multiplication Rule | an * bn = (a*b)n | an * am = a(m+n) |
| Division Rule | an/ bn = (a/b)n | am / an = a(m-n) |
| Power Rule | (an)m = (a)nm | n√a = a(1/n) |
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.
Questions and Answers based on Formulas
Question 1:
Find the value of (3x+2y)² using (a+b)² formula.
Solution:
To find: (3x+2y)²
Question 2:
Solve the following expression using suitable algebraic identity: (2x+3y)³
Solution:
To find: (2x+3y)³