NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.4

Welcome to your YoLearn AI comprehensive study guide for Exercise 1.4 of CBSE Class 9 Mathematics. This crucial section of Chapter 1 focuses on performing mathematical operations on real numbers. You will discover how rational and irrational numbers interact under addition, subtraction, multiplication, and division. Crucially, this exercise introduces you to algebraic identities involving square roots and teaches you the indispensable technique of 'rationalizing the denominator'—a fundamental skill that you will rely on throughout high school algebra and trigonometry. Let's master these operations step-by-step, ensuring you can solve your exam problems with complete confidence!

Key Algebraic Rules for Real Numbers

When we operate on real numbers, we follow specific algebraic rules. When combining rational and irrational numbers, remember:

  1. The sum or difference of a rational number and an irrational number is always irrational.
  2. The product or quotient of a non-zero rational number with an irrational number is always irrational.
  3. If we add, subtract, multiply, or divide two irrationals, the result may be rational or irrational.

To simplify expressions containing square roots, we apply the following radical identities for positive real numbers 'a' and 'b':

  • Rule 1: √ab = √a * √b
  • Rule 2: √(a/b) = √a / √b
  • Rule 3: (√a + √b)(√a - √b) = a - b
  • Rule 4: (a + √b)(a - √b) = a^2 - b
  • Rule 5: (√a + √b)^2 = a + 2√ab + b

Using these standard formulas allows us to systematically expand and simplify complex algebraic expressions.

The Step-by-Step Denominator Rationalization Method

  1. Identify the Denominator's Structure — Examine the denominator of the fraction. If it contains irrational terms, identify whether it is a single radical (e.g., √a) or a binomial containing a radical (e.g., a + √b).
  2. Determine the Rationalizing Factor (Conjugate) — Find the factor that will eliminate the square root. For a simple denominator √a, the rationalizing factor is √a. For a binomial denominator like (a + √b), its conjugate is (a - √b). If the denominator is (a - √b), the conjugate is (a + √b).
  3. Multiply Numerator and Denominator — Multiply both the top and the bottom of the fraction by the identified conjugate factor. This preserves the overall value of the fraction.
  4. Simplify and Remove Radicals from the Denominator — Simplify the numerator. In the denominator, apply the difference of squares identity: (x - y)(x + y) = x^2 - y^2 to transform the irrational denominator into a rational integer.

Solved Exemplar Class 9 Math Problems

  • Example 1: Simplify the expression (5 + √5)(5 - √5). Step 1: Identify the matching algebraic identity. This fits the form (a + √b)(a - √b) = a^2 - b. Step 2: Substitute the values into the formula: a = 5 and b = 5. Step 3: Calculate the result: (5)^2 - 5 = 25 - 5 = 20. Final Answer: 20
  • Example 2: Rationalize the denominator of 1 / (2 + √3). Step 1: Identify the conjugate of the denominator. The conjugate of 2 + √3 is 2 - √3. Step 2: Multiply the numerator and denominator by the conjugate: [1 * (2 - √3)] / [(2 + √3)(2 - √3)]. Step 3: Simplify the denominator using (a + b)(a - b) = a^2 - b^2: (2)^2 - (√3)^2 = 4 - 3 = 1. Step 4: Write the final simplified fraction: (2 - √3) / 1 = 2 - √3. Final Answer: 2 - √3

Board Exam Pro-Tips & Common Student Mistakes

Avoid these typical errors during exams:

  • Incorrect Identity Fallacy: Never write √(a + b) = √a + √b. For instance, √(9 + 16) is √25 = 5, whereas √9 + √16 = 3 + 4 = 7. They are not equal!
  • Conjugate Sign Errors: While rationalizing, remember to flip the operational sign. If the denominator is (5 - √2), multiplying by (5 - √2) will not rationalize it; you must multiply by its conjugate (5 + √2).
  • Forgetting to Distribute: When multiplying binomials containing radicals, ensure you distribute every term using the FOIL method (First, Outer, Inner, Last).

Practice Questions with Solutions

  • Q: Classify the number (3 + √23) - √23 as rational or irrational. A: Step 1: Open the brackets to write the expression as: 3 + √23 - √23. Step 2: Simplify the expression by cancelling the equal and opposite radical terms: √23 - √23 = 0. Step 3: The expression reduces to 3. Final answer: Since 3 is an integer, it can be expressed as 3/1. Therefore, (3 + √23) - √23 is a Rational Number.
  • Q: Simplify the expression (√3 + √2)^2. A: Step 1: Identify the identity: (x + y)^2 = x^2 + 2xy + y^2. Step 2: Substitute x = √3 and y = √2. Step 3: Expand the expression: (√3)^2 + 2(√3)(√2) + (√2)^2. Step 4: Calculate individual terms: 3 + 2√6 + 2. Step 5: Combine rational terms: 5 + 2√6. Final answer: 5 + 2√6
  • Q: Rationalize the denominator of 5 / (√3 - √5). A: Step 1: Identify the conjugate of (√3 - √5), which is (√3 + √5). Step 2: Multiply the numerator and denominator: [5 * (√3 + √5)] / [(√3 - √5)(√3 + √5)]. Step 3: Apply the identity (√a - √b)(√a + √b) = a - b in the denominator: 3 - 5 = -2. Step 4: Simplify the expression: 5(√3 + √5) / -2 = -5(√3 + √5) / 2. Final answer: -5(√3 + √5) / 2
  • Q: Rationalize the denominator of 1 / √7. A: Step 1: Identify the rationalizing factor for √7, which is √7. Step 2: Multiply both top and bottom by √7: (1 √7) / (√7 √7). Step 3: Simplify the denominator: √7 * √7 = 7. Final answer: √7 / 7

Frequently Asked Questions

What does it mean to rationalize the denominator?

Rationalizing the denominator means converting an irrational number in the denominator of a fraction into a rational number. This is achieved by multiplying the numerator and denominator by an appropriate conjugate factor that eliminates the radical.

How do you find the conjugate of a radical expression?

To find the conjugate of a two-term radical expression, simply reverse the sign connecting the two terms. For instance, the conjugate of (x + √y) is (x - √y), and the conjugate of (√a - √b) is (√a + √b).

Is the product of two irrational numbers always irrational?

No, the product of two irrational numbers can be rational or irrational. For example, √2 multiplied by √3 is √6 (irrational), but √2 multiplied by √2 is 2 (rational).