📘 Exercise 1.2
Learn and practice Class 9 Mathematics Exercise 1.2 on Real Numbers with step-by-step solutions. This page includes free downloadable notes and solved examples aligned with the PECTAA 2025 syllabus.
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Chapter 1
Real Numbers
Chapter 2
Logarithms
Chapter 3
Sets and Functions
Chapter 4
Factorization and Algebraic Manipulation
Chapter 5
Linear Equations and Inequalities
Chapter 6
Trigonometry
Chapter 7
Coordinate Geometry
Chapter 8
Logic
Chapter 9
Similar Figures
Chapter 10
Graphs of Functions
Chapter 11
Loci and Construction
Chapter 12
Information Handling
Chapter 13
Probability
Exercise 1.2 - Real Numbers
This exercise focuses on rationalizing denominators, simplifying expressions with exponents, and working with algebraic identities.
Exercise 1.2 provides practice problems that help students master the technique of rationalizing denominators, simplify complex expressions using laws of exponents, and apply algebraic identities to solve problems.
The exercise covers rationalizing denominators with single and binomial surds, simplifying expressions with fractional exponents, and applying algebraic identities to find values of expressions.
Key Topics Covered
- Rationalizing denominators with single surds
- Rationalizing denominators with binomial surds
- Simplifying expressions with exponents
- Applying laws of exponents
- Using algebraic identities
- Finding values of expressions using given conditions
Important Note:
Remember that rationalizing the denominator means eliminating radicals from the denominator of a fraction. For binomial denominators containing surds, we multiply numerator and denominator by the conjugate of the denominator.
Important Formulas Used in This Exercise
Square Identities
$(x+y)^2 = x^2 + 2xy + y^2$
$(x-y)^2 = x^2 - 2xy + y^2$
$x^2 - y^2 = (x+y)(x-y)$
Cube Identities
$x^3 + y^3 = (x+y)(x^2 - xy + y^2)$
$x^3 - y^3 = (x-y)(x^2 + xy + y^2)$
Exponent Rules
$a^m \cdot a^n = a^{m+n}$
$(a^m)^n = a^{mn}$
$\frac{a^m}{a^n} = a^{m-n}$
$a^0 = 1$
Sample Problems with Solutions
Problem 1: Rationalize the Denominator
Rationalize the denominator of $\frac{13}{4+\sqrt{3}}$
1
Multiply numerator and denominator by the conjugate
$\frac{13}{4+\sqrt{3}} \times \frac{4-\sqrt{3}}{4-\sqrt{3}}$
2
Apply the formula $(a+b)(a-b) = a^2 - b^2$
$\frac{13(4-\sqrt{3})}{(4)^2-(\sqrt{3})^2} = \frac{13(4-\sqrt{3})}{16-3}$
3
Simplify
$\frac{13(4-\sqrt{3})}{13} = 4-\sqrt{3}$
Final Answer: $4-\sqrt{3}$
Problem 2: Simplify Exponential Expression
Simplify $\left(\frac{81}{16}\right)^{-\frac{3}{4}}$
1
Apply negative exponent rule
$\left(\frac{81}{16}\right)^{-\frac{3}{4}} = \left(\frac{16}{81}\right)^{\frac{3}{4}}$
2
Express numbers as powers
$\left(\frac{2^4}{3^4}\right)^{\frac{3}{4}} = \frac{(2^4)^{\frac{3}{4}}}{(3^4)^{\frac{3}{4}}}$
3
Apply power of a power rule
$\frac{2^{4 \cdot \frac{3}{4}}}{3^{4 \cdot \frac{3}{4}}} = \frac{2^3}{3^3} = \frac{8}{27}$
Final Answer: $\frac{8}{27}$
Created by Hira Science Academy | Aligned with PECTA 2025 Syllabus