Exercise 1.4 — Operations on Real Numbers

Operations on Real Numbers

Before diving into Exercise 1.4, it helps to understand how rational and irrational numbers behave under basic arithmetic. Rational numbers are closed under addition, subtraction, and multiplication — the result of any such operation between two rationals is always rational. However, irrational numbers are not closed under these operations. For example, adding √5 and −√5 gives 0, which is rational, and √11 × √11 = 11, also rational. This is an important distinction tested in CBSE, Telangana, and Andhra Pradesh board exams.

A key rule to remember: when a rational number is combined with an irrational number through addition, subtraction, multiplication, or division, the result is always irrational. For instance, 5 + √3, 5 − √3, 5√3, and 5/√3 are all irrational.

Useful Identities for Simplifying Expressions

Exercise 1.4 relies heavily on algebraic identities applied to surds. These identities allow you to expand, simplify, and rationalise expressions involving square roots:

• √(ab) = √a · √b — Product rule for square roots of non-negative numbers.
• (√a + √b)(√a − √b) = a − b — Difference of squares; this is the basis of rationalisation.
• (√a + √b)² = a + 2√(ab) + b — Expansion of a binomial surd squared.
• (√a + √b)(√c + √d) = √(ac) + √(ad) + √(bc) + √(bd) — FOIL method applied to surds.

Worked example from Q1(i): Simplify (√5 + √7)(2 + √5). Expanding term by term gives 2√5 + 5 + 2√7 + √35. So the answer is 10 + 5√5 + 2√7 + √35. For Q1(ii), (√5 + √5)(√5 − √5) uses the identity (a + b)(a − b) = a² − b, giving 25 − 5 = 20.

Rationalising the Denominator

A rationalising factor (R.F.) of an irrational number is another irrational number such that their product is rational. For example, √3 × √27 = √81 = 9, so √3 and √27 are rationalising factors of each other. Note that the R.F. of a number is not unique — it is always best to use the simplest one.

For expressions of the form (√a + √b) or (√a − √b), the R.F. is its conjugate: (√a − √b) or (√a + √b) respectively, since their product gives a − b, a rational number. This technique is used throughout Questions 5 and 6.

Worked example from Q5(i): To rationalise 1/(√3 + √2), multiply numerator and denominator by (√3 − √2). The denominator becomes 3 − 2 = 1, giving the simplified result (√3 − √2)/7... wait — since (√3)² − (√2)² = 3 − 2 = 1, the answer is simply (√3 − √2). Applying the same method to Q5(ii): 1/(√7 − √6) × (√7 + √6)/(√7 + √6) = (√7 + √6)/(7 − 6) = √7 + √6.

Surds and Exponential Form

A surd is the positive nth root of a positive rational number that is not itself a perfect nth power. Formally, ⁿ√a (written as a^1/n) is a surd when 'a' is a positive rational but not the nth power of any rational number. The number under the radical sign is called the radicand, the symbol √ is the radical sign, and 'n' is the degree of the radical.

• Examples of surds: √5, ∛4, ⁴√7, ∛9 — none of these simplify to a rational number.
• Non-surds: √9 = 3, ∛27 = 3, ⁴√16 = 2 — these are perfect powers and evaluate to rationals.
• Exponential form: ⁿ√a = a^1/n. For example, ∛9 = 9^1/3.

Questions 8 and 9 test the laws of exponents extended to rational exponents. For instance, 64^1/6 = (2⁶)^1/6 = 2¹ = 2. The simplification in Q9 — 4√81 − 8·∛343 + 15·⁵√32 + √225 — works out to 3 − 56 + 30 + 15 = −8 after evaluating each root individually.

Common Mistakes to Avoid

• Assuming irrational numbers are always closed under operations — they are not. √5 + (−√5) = 0 is rational.
• Forgetting to multiply both numerator and denominator by the rationalising factor, which changes the value of the expression.
• Using the wrong conjugate — the R.F. of (√a + √b) is (√a − √b), not (−√a + √b).
• Confusing surds with all irrationals — π is irrational but not a surd, because it is not expressible as a root of a rational number.
• Errors in applying (√a + √b)² — students often write a² + b instead of a + 2√(ab) + b.

What This Exercise Prepares You For

Mastering Exercise 1.4 builds the foundation for working with irrational expressions across many chapters. The rationalisation technique is directly applied when simplifying trigonometric ratios and coordinate geometry expressions in later classes. Strong understanding of surds and exponent laws also makes topics like irrational numbers and Class 10 Real Numbers much easier to approach. Students preparing for CBSE, Telangana, or Andhra Pradesh board exams should be confident in classifying numbers as rational or irrational (Q2, Q3) and in finding unknown rational values by comparing coefficients (Q10).