NCERT Solutions Ganita Prakash (Part 1) Chapter 2 .2 Exponential Notation and Operations — In-text Questions

Book page 222 Updated on2026-09-05

Q1.

Which expression describes the thickness of a sheet of paper after it is folded 10 times? The initial thickness is represented by the letter-number v. (i) 10v (ii) 10 + v (iii) 2 × 10 × v (iv) 2¹⁰ (v) 2¹⁰v (vi) 10²v

Answer

(v) 2¹⁰v

Start: v
After 1 fold: v × 2
After 2 folds: v × 2 × 2 = v × 2²
…
After 10 folds: v × 2 × 2 × … × 2 (ten 2s) = 2¹⁰v = 1024v

Why the others fail:

10v and 2 × 10 × v multiply by a fixed number of times folded, not by 2 each time — that is linear thinking.
10 + v adds; folding never adds.
2¹⁰ forgets the starting thickness altogether — it is a plain number, not a thickness.
10²v = 100v; but the base must be 2 (the doubling), and the exponent must be 10 (the number of folds). Here they have been swapped.

Why it happens: in n^a, the base n is what is repeated and the exponent a is how many times. Folding repeats "× 2" ten times, so the base is 2 and the exponent is 10 — never the other way round.

Q2.

Express the number 32400 as a product of its prime factors and represent the prime factors in their exponential form.

Answer

Divide repeatedly by the smallest prime that goes in, as in the factor tree on this page.

32400 ÷ 2 = 16200
16200 ÷ 2 = 8100
8100 ÷ 2 = 4050
4050 ÷ 2 = 2025
2025 ÷ 5 = 405
405 ÷ 5 = 81
81 ÷ 3 = 27 ÷ 3 = 9 ÷ 3 = 3 ÷ 3 = 1

32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3
= 2⁴ × 5² × 3⁴

Check: 2⁴ = 16, 5² = 25, 3⁴ = 81, and 16 × 25 × 81 = 400 × 81 = 32400. ✓

Why it happens: exponential form is not just shorter — it is the form in which the number's structure is visible. Reading 2⁴ × 5² × 3⁴ you can see at once that 32400 is a perfect square (every exponent is even), that it is divisible by 16, by 25 and by 81, and that its square root is 2² × 5 × 3² = 180.

Q3.

What is (– 1)⁵ ? Is it positive or negative? What about (– 1)⁵⁶ ?

Answer

(–1)⁵ = (–1)(–1)(–1)(–1)(–1)
= [(–1)(–1)] × [(–1)(–1)] × (–1)
= 1 × 1 × (–1) = –1 (negative)

(–1)⁵⁶ = 28 pairs of (–1)(–1)
= 1 × 1 × … × 1 = 1 (positive)

Why it happens: the minus signs cancel two at a time, because (–1) × (–1) = +1. So only the parity of the exponent matters. An even power of a negative number is positive (all the minus signs pair up); an odd power is negative (one minus sign is left over). Here 5 is odd and 56 is even.

Tip: this works for any negative base: (–3)⁴ = 81 but (–3)³ = –27. Be careful with brackets, though — (–3)² = 9 while –3² = –9, because in the second one only the 3 is squared.

Q4.

What is 0², 0⁵ ? What is 0ⁿ ?

Answer

0² = 0 × 0 = 0
0⁵ = 0 × 0 × 0 × 0 × 0 = 0
0ⁿ = 0, for every counting number n

Why it happens: a product is zero the moment one factor is zero. Multiply 0 by itself any number of times and there is nothing to build up with — unlike doubling, which grows, or halving, which shrinks but never dies, repeated multiplication by 0 collapses at the first step and stays there.

Did you know? The one case left out is 0⁰. The rule n⁰ = 1 needs n ≠ 0, and the pattern 0ⁿ = 0 would want the answer to be 0. The two patterns disagree, so 0⁰ is left undefined. This is also the reason the chapter insists n ≠ 0 in n^a ÷ n^b.

Q5.

Is (– 2)⁴ = 16? Verify.

Answer

Yes.

(–2)⁴ = (–2) × (–2) × (–2) × (–2)
= [(–2) × (–2)] × [(–2) × (–2)]
= 4 × 4
= 16

Why it happens: 4 is even, so the four minus signs form two pairs and every pair gives +1. In general (–n)^a = n^a when a is even, and (–n)^a = –n^a when a is odd. Compare with the previous question: (–2)³ = –8, not 8.

Was this helpful? Report an error