Basic Algebraic Identities of School Algebra and Elementary Surds
What is Basic Algebraic Identities of School Algebra and Elementary Surds?
An equation that is true for all values of the variables involved. It provides a shortcut for expanding or factoring expressions.
Key formula / rule: Square of a sum
Key points
- To simplify algebraic expressions using identities.
- To factorize expressions using identities.
- To solve equations involving surds.
- To rationalize denominators of expressions containing surds.
Common exam trap
Confusing (a+b)² with a²+b²
Definitions
- Term
Algebraic Identity
- Meaning
An equation that is true for all values of the variables involved. It provides a shortcut for expanding or factoring expressions.
- Term
Surd
- Meaning
An irrational root of a rational number, typically expressed using the radical symbol (e.g., √2, ³√5). Elementary surds usually refer to square roots.
- Term
Rationalizing the Denominator
- Meaning
The process of transforming a fraction with a surd in the denominator into an equivalent fraction with a rational number in the denominator.
- Term
Conjugate
- Meaning
For an expression of the form a + √b, its conjugate is a - √b. Multiplying an expression by its conjugate helps in rationalizing the denominator.
Learning objectives
To simplify algebraic expressions using identities.
To factorize expressions using identities.
To solve equations involving surds.
To rationalize denominators of expressions containing surds.
To efficiently compute values of complex expressions.
Formulae
- Name
Square of a sum
- Note
Used for expanding binomials and simplifying expressions.
- Expression
(a+b)² = a² + 2ab + b²
- Name
Square of a difference
- Note
Used for expanding binomials and simplifying expressions.
- Expression
(a-b)² = a² - 2ab + b²
- Name
Difference of squares
- Note
Used for factorization and simplification.
- Expression
a² - b² = (a+b)(a-b)
- Name
Sum of cubes
- Note
Used for factorization.
- Expression
a³ + b³ = (a+b)(a² - ab + b²)
- Name
Difference of cubes
- Note
Used for factorization.
- Expression
a³ - b³ = (a-b)(a² + ab + b²)
- Name
Rationalizing factor
- Note
Used to remove square roots from the denominator.
- Expression
For a+√b, the rationalizing factor is a-√b. For √a+√b, it is √a-√b.
Prerequisites
Basic arithmetic operations
Understanding of variables and constants
Familiarity with exponents and powers
Common mistakes
Confusing (a+b)² with a²+b²
Incorrectly applying the difference of squares formula
Errors in sign during expansion or factorization
Mistakes in rationalizing the denominator (e.g., wrong conjugate)
Assuming √a + √b = √(a+b)
Keywords
Algebraic Identities
Surds
Square Roots
Rationalization
Binomial Expansion
Factorization
Simplification
Quantitative Aptitude
SSC CGL
Practice preview
If (a + b) = 12 and ab = 35, what is the value of a^2 + b^2?…
easy
Which of the following is equivalent to (a + b + c)^2 - (a^2 + b^2 + c^2)?…
easy
Simplify the expression by rationalizing the denominator: 1 / (sqrt(5) - sqrt(3)).…
medium