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NCERT Solutions Ganita Prakash (Part 1) Chapter 6 .1 Some Properties of Multiplication — In-text Questions

Book page 1416 Updated on2026-09-05

Q1.
Can any two terms be added to get a single term? For example, can 3⁄2 a² and 3⁄10 a be added to get a single term?
Answer

No. Two terms can be combined into one only when they are like terms — that is, when they have exactly the same letter-numbers.

3⁄2 a² has the letter part a × a
3⁄10 a has the letter part a
a × a ≠ a, so they are unlike terms

Test at a = 4: 3⁄2 × 16 = 24 and 3⁄10 × 4 = 1.2, giving 25.2. There is no single number k with k a² = 25.2 or k a = 25.2 for every a — at a = 10 the sum would be 153, not 25.2 scaled.

Why it happens: adding like terms works because of distributivity read backwards: 2ab + 3ab = (2 + 3) ab. You can only pull a common factor out if both terms actually contain it. a² and a share the factor a, so 3⁄2 a² + 3⁄10 a = a(3⁄2 a + 3⁄10) — but that is a product of two things, not a single term.
Q2.
Example 2: Expand (a + b) (a + b).
Answer
(a + b)(a + b) = (a + b) a + (a + b) b
= a × a + b × a + ab + b × b
= a² + ba + ab + b²

Since ba = ab, the middle two are like terms:

ba + ab = ab + ab = 2ab
(a + b)(a + b) = a² + 2ab + b²

Test at a = 3, b = 5: LHS = 8 × 8 = 64; RHS = 9 + 30 + 25 = 64 ✓

Why it happens: the four products are a·a, a·b, b·a and b·b. Two of them are the same because multiplication is commutative — that single fact is the whole reason a square of a sum has a “2ab” in the middle rather than two separate terms.
Q3.
Example 3: Expand (a + b) (a² + 2ab + b²).
Answer
(a + b)(a² + 2ab + b²) = (a + b)a² + (a + b) × 2ab + (a + b)b²
= (a × a²) + ba² + (a × 2ab) + (b × 2ab) + ab² + (b × b²)

Simplify each term:

a × a² = a³ (a × a × a)
ba² = a²b
a × 2ab = 2 × a × a × b = 2a²b
b × 2ab = 2 × a × b × b = 2ab²
b × b² = b³
= a³ + a²b + 2a²b + 2ab² + ab² + b³

Now collect the like terms:

a²b + 2a²b = (1 + 2) a²b = 3a²b
ab² + 2ab² = (1 + 2) ab² = 3ab²
(a + b)(a² + 2ab + b²) = a³ + 3a²b + 3ab² + b³

Test at a = 2, b = 1: LHS = 3 × (4 + 4 + 1) = 27; RHS = 8 + 12 + 6 + 1 = 27 ✓

Did you know? Since a² + 2ab + b² is itself (a + b)², this shows (a + b)³ = a³ + 3a²b + 3ab² + b³. The coefficients 1, 3, 3, 1 are the next row after 1, 2, 1.
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