Exercise 1.2 Class 9 Math Solutions – Real Numbers

If you’re looking for Exercise 1.2 Class 9 Math notes, you’re in the right place! This exercise from Unit 1: Real Numbers focuses on important concepts such as rationalizing denominators, simplifying expressions with exponents and radicals, and using algebraic identities effectively.

Many students find these topics tricky at first, so these step-by-step solutions are written just like a teacher would explain them in class. The notes are prepared by experienced educators following the latest Class 9 Mathematics syllabus in Pakistan.

You can also download the PDF for free — perfect for practicing at home, revising before exams, or checking your answers. All solutions are carefully reviewed for accuracy and clarity, helping you build confidence for Chapter 1.

Download/ View Notes of Exercise 1.2

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Common Mistakes to Avoid in Exercise 1.2 Class 9 Math

1. Incorrect Application of Exponent Rules

Rule:
aᵐ × aⁿ = aᵐ⁺ⁿ
This rule only applies when the base is the same.
Mistake Example:
2³ × 3² ≠ 6⁵
The bases are different (2 and 3), so you cannot add the exponents.

2. Partial Simplification

After rationalizing a denominator, make sure you simplify the expression completely.
Mistake Example:
\( \frac{52 – 13\sqrt{3}}{13} \rightarrow 4 – \sqrt{3} \)
Don’t stop at the unsimplified form. Divide each term in the numerator by 13 to simplify fully.

3. Ignoring Conjugate Pairs

When rationalizing expressions with square roots, students sometimes fail to multiply all terms in the numerator by the conjugate of the denominator.
Tip:
If the denominator is a + √b, multiply both numerator and denominator by a − √b, and vice versa. Be sure to apply multiplication to every term in the numerator.

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Key Definitions

Surd:

An irrational number expressed as a non-terminating, non-repeating root.
Examples: \( \sqrt{2}, \sqrt{3}, \sqrt{5} \)

Conjugate:

A pair of expressions like \( (a + \sqrt{b}) ) and ( (a – \sqrt{b}) \) used to rationalize denominators.

Rational Number:

A number that can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \ne 0 \).

Indices:

The plural of index (or exponent). It tells how many times a number (called the base) is multiplied by itself.
Example: \( a^m \) means \( a \times a \times a \dots \) \(m–times\).

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Important Laws and Formulas

Laws of Radicals	Laws of Indices
\( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \) \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \) \( \sqrt[n]{a^m} = \left( \sqrt[n]{a} \right)^m \) \( \left( \sqrt[n]{a} \right)^{1/n} = \left( a^n \right)^{1/n} = a \)	\( a^m \cdot a^n = a^{m+n} \) \( (a^m)^n = a^{mn} \) \( (ab)^n = a^n b^n \) \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \) \( \frac{a^m}{a^n} = a^{m-n} \) \( a^0 = 1 \)

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📐 Real-Life Applications

Surds in Architecture & Carpentry:

• Calculating the exact length of rafters or beams when building a roof (using √2 for 45° angles).
• Measuring the diagonal length of rectangular tiles or screens (for example, TV screen size measured diagonally involves √(width² + height²)).

Radicals in Medicine & Biology:

• Calculating dosages or growth rates that involve square roots, such as in pharmacokinetics or modeling population growth.
• Estimating the surface area of organs or tumors where radical formulas appear in volume and area calculations.

Exponents in Computer Science & Technology:

• Understanding how data storage grows exponentially (e.g., bits and bytes doubling with each memory upgrade).
• Algorithms involving powers and roots for encryption, data compression, or signal processing.

Exponents in Finance & Economics:

Modeling population growth or inflation rates which follow exponential growth or decay. Calculating compound interest on savings or loans where the formula is A = P × (1 + r)ⁿ.

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Short Questions

Question 1: Define a surd. Give an example of a binomial surd.

Answer: A surd is an irrational number expressed as a root (e.g., \( \sqrt{5} )\). A binomial surd contains two terms, such as \( 1 + \sqrt{3} \).

Question 2: Rationalize \( \frac{7}{\sqrt{5} + \sqrt{3}} \).
Answer: Multiply numerator and denominator by the conjugate \( \sqrt{5} – \sqrt{3} \):
\[
\frac{7}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} – \sqrt{3}}{\sqrt{5} – \sqrt{3}} = \frac{7(\sqrt{5} – \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} – \sqrt{3})} = \frac{7(\sqrt{5} – \sqrt{3})}{5 – 3} = \frac{7(\sqrt{5} – \sqrt{3})}{2}
\]
Question 3: Simplify \( \left( \frac{27}{125} \right)^{-\frac{2}{3}} \).
Answer: Rewrite bases as powers:
\[
\left( \frac{3^{3}}{5^{3}} \right)^{-\frac{2}{3}} = \frac{5^{2}}{3^{2}} = \frac{25}{9}
\]
Question 4: Expand \( (3 + 2)(3 – 2) \). What algebraic identity does this demonstrate?
Answer:
(3 + 2)(3 − 2) = 3² − 2² = 9 − 4 = 5
Identity used: (a + b)(a − b) = a² − b².

Question 5: Explain why \( \sqrt{5} \times \sqrt{5} \) is rational, but \( 2 + \sqrt{3} \) is irrational.
Answer: \( \sqrt{5} \times \sqrt{5} = 5 \), which is rational.
\( 2 + \sqrt{3} \) cannot be simplified to a fraction and remains irrational.

Question 6: Simplify Simplify 3ⁿ⁺² × 9ⁿ⁻¹.
Answer: Rewrite 9 as 3²:
3ⁿ⁺² × (3²)ⁿ⁻¹ = 3ⁿ⁺² × 3²ⁿ⁻² = 3³ⁿ
Question 7: If \( x = 2 + \sqrt{5} \), find \( x^{2} + \frac{1}{x^{2}} \).
Answer:
\[
x^{2} = (2 + \sqrt{5})^{2} = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5}
\]
\[
\frac{1}{x} = \frac{1}{2 + \sqrt{5}} \times \frac{2 – \sqrt{5}}{2 – \sqrt{5}} = 2 – \sqrt{5}
\]
\[
\Rightarrow \frac{1}{x^{2}} = (2 – \sqrt{5})^{2} = 4 – 4\sqrt{5} + 5 = 9 – 4\sqrt{5}
\]
Sum:
\[
x^{2} + \frac{1}{x^{2}} = (9 + 4\sqrt{5}) + (9 – 4\sqrt{5}) = 18
\]
Question 8: Calculate the diagonal length of a square with side \( 5\sqrt{2} \) cm. Express your answer as a surd.
Answer:
Diagonal length \(= \text{side} \times \sqrt{2} = 5\sqrt{2} \times \sqrt{2} = 5 \times 2 = 10 \) cm.

We hope these 9 class math notes for Exercise 1.2 helped you understand the important concepts of real numbers, including rationalization and simplification. To keep improving, don’t stop here—explore more exercises from Chapter 1 to strengthen your foundation:

• ✅ Exercise 1.1 – Real Numbers | Solved Notes PDF
• ✅ Exercise 1.3 – Real Numbers | Solved Notes PDF

Also, if you need the full book for reference or revision, download it here:

• 📘 Class 9 Math Textbook – Punjab Board PDF

Keep practicing and stay confident. If you find these notes helpful, share them with your friends and continue learning with notesofmath.com!

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The Class 9 Mathematics book is the property of the Punjab Curriculum and Textbook Board (PCTB). NotesofMath.com provides these solved exercises only for educational and revision purposes to help students better understand the syllabus concepts. As this is a newly introduced book, some printing or conceptual errors may exist in the original text. For official curriculum and textbook information, please visit the PCTB website.