NCERT Solutions Class 9 Maths Number Systems Ex 1.5

Welcome to the study guide for CBSE Class 9 Maths Chapter 1, Exercise 1.5! In this section of Number Systems, we dive deep into the fascinating world of operations on real numbers. Up until now, you have studied rational and irrational numbers as separate entities. Now, you will learn what happens when we add, subtract, multiply, or divide them. Exercise 1.5 introduces critical mathematical concepts such as the algebraic properties of square roots and the highly important process of 'rationalizing the denominator'. Rationalization is a foundational algebraic tool that you will use throughout your high school mathematics, especially in trigonometry and calculus. By mastering this exercise, you will build a solid foundation for evaluating complex irrational expressions and solving textbook problems with speed and precision. Let's explore the core formulas, step-by-step methods, and key practice problems with our YoLearn AI sketchpad approach to make this chapter effortless!

Understanding Operations on Real Numbers

When we perform mathematical operations on real numbers, some unique rules apply. If we add, subtract, multiply, or divide two rational numbers, we always get a rational number. However, operations involving irrational numbers behave differently. The sum or difference of a rational number and an irrational number is always irrational (for example, 3 + √2 is irrational). Similarly, the product or quotient of a non-zero rational number with an irrational number is always irrational (like 3√2). But what happens when we operate on two irrational numbers? Interestingly, the sum, difference, product, or quotient of two irrational numbers may be rational or irrational. For instance, √3 multiplied by √3 equals 3, which is rational, whereas √2 multiplied by √3 equals √6, which is irrational. Understanding these combinations is essential for solving the simplifications in Exercise 1.5.

Key Algebraic Identities for Radicals

Square Root Identity 1
√(ab) = √a * √b, where a and b are positive real numbers.
Square Root Identity 2
√(a/b) = √a / √b, where a and b are positive real numbers and b is non-zero.
Conjugate
The conjugate of (√a + √b) is (√a - √b). Multiplying a binomial irrational expression by its conjugate eliminates the radical terms, utilizing the difference of squares algebraic identity: (x + y)(x - y) = x² - y².

How to Rationalize the Denominator

  1. Identify the Denominator — Locate the irrational part in the denominator of the fraction, such as 1 / (√a + b).
  2. Find the Conjugate — Change the mathematical sign between the terms of the denominator to find its conjugate. For (√a + b), the conjugate is (√a - b).
  3. Multiply Numerator and Denominator — Multiply both the top and the bottom of the fraction by this conjugate. This maintains the fraction's equivalent value.
  4. Simplify the Expression — Apply the identity (x + y)(x - y) = x² - y² to simplify the denominator. This eliminates the radical terms in the denominator, leaving a clean rational number.

Common Pitfalls to Avoid in Board Exams

Many students make the mistake of distributing square roots over addition, writing √(a + b) = √a + √b. This is strictly incorrect. For example, √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Always treat radical expressions like variables: you can only add or subtract 'like surds' (such as 2√3 + 5√3 = 7√3). Another common mistake is failing to multiply both the numerator and denominator by the conjugate when rationalizing; doing it to only one side completely alters the mathematical value of the expression.

Practice Questions with Solutions

  • Q: Classify the following numbers as rational or irrational: (i) 2 - √5, (ii) (3 + √23) - √23. A: Step 1: Analyze expression (i): 2 - √5. Here, 2 is a rational number and √5 is an irrational number. The difference between a rational and an irrational number is always irrational. Therefore, 2 - √5 is irrational. Step 2: Simplify expression (ii): (3 + √23) - √23. Remove brackets to get 3 + √23 - √23. The +√23 and -√23 cancel each other out, leaving only 3. Since 3 can be written as 3/1, it is a rational number. Final answer: (i) Irrational, (ii) Rational.
  • Q: Simplify the expression: (3 + √3)(2 + √2). A: Step 1: Use the distributive property of multiplication (FOIL method) to expand the brackets: (a + b)(c + d) = ac + ad + bc + bd. Step 2: Apply this to the given expression: First term multiplication: 3 2 = 6. Outer term multiplication: 3 √2 = 3√2. Inner term multiplication: √3 2 = 2√3. Last term multiplication: √3 √2 = √(3 * 2) = √6. Step 3: Combine all the simplified terms: 6 + 3√2 + 2√3 + √6. Since none of these are like surds, we cannot simplify further. Final answer: 6 + 3√2 + 2√3 + √6.
  • Q: Rationalize the denominator of: 1 / (√7 - √6). A: Step 1: Identify the denominator, which is (√7 - √6). Its conjugate is (√7 + √6). Step 2: Multiply both the numerator and denominator of the fraction by the conjugate: [1 (√7 + √6)] / [(√7 - √6) (√7 + √6)]. Step 3: Simplify the numerator: 1 * (√7 + √6) = √7 + √6. Step 4: Simplify the denominator using the identity (a - b)(a + b) = a² - b²: (√7)² - (√6)² = 7 - 6 = 1. Step 5: Write the complete simplified fraction: (√7 + √6) / 1 = √7 + √6. Final answer: √7 + √6.
  • Q: Rationalize the denominator of: 5 / (√3 - √5). A: Step 1: Identify the denominator, which is (√3 - √5). Its conjugate is (√3 + √5). Step 2: Multiply both the numerator and the denominator by (√3 + √5): [5 (√3 + √5)] / [(√3 - √5) (√3 + √5)]. Step 3: Expand the numerator: 5(√3 + √5) = 5√3 + 5√5. Step 4: Simplify the denominator using the identity (a - b)(a + b) = a² - b²: (√3)² - (√5)² = 3 - 5 = -2. Step 5: Assemble the fraction and place the negative sign cleanly: [5(√3 + √5)] / -2 = -5(√3 + √5) / 2. Final answer: -5(√3 + √5) / 2.

Frequently Asked Questions

What does it mean to rationalize the denominator?

Rationalizing the denominator means converting an irrational denominator into a rational number. This is done by multiplying both the numerator and denominator by an appropriate conjugate, making the fraction easier to evaluate.

Can the product of two irrational numbers ever be rational?

Yes, the product of two irrational numbers can be rational. For example, multiplying √2 by itself gives √4, which simplifies directly to the rational number 2.

How do we identify the conjugate of a binomial denominator containing square roots?

To find the conjugate of a two-term expression containing square roots, you simply reverse the mathematical sign between the terms. For instance, the conjugate of a + √b is a - √b.