NCERT Solutions Ganita Prakash (Part 1) Chapter 2 .3 The Other Side of Powers · Power Lines — In-text Questions
Book page 302 Updated on2026-09-05
Q1.
How many times larger than 4⁻² is 4²?
Answer
44 = 256 times.
42 ÷ 4–2 = 42–(–2) = 42+2 = 44 = 256
Check with the values on the power line: 42 = 16 and 4–2 = 1/16, so 16 ÷ (1/16) = 16 × 16 = 256. ✓
Why it happens: on the power line the two numbers sit four steps apart — 4–2, 4–1, 40, 41, 42 — and each step multiplies by 4. Four steps therefore multiply by 44. The power line turns "how many times larger" into "how many steps apart", which is the whole point of arranging powers along a line: a ratio of values becomes a difference of exponents.
Q2.
Use the power line for 7 to answer the following questions. 2,401 × 49 = 49³ = 343 × 2,401 = 16,807/49 = 7/343 = 16,807/8,23,543 = 1,17,649 × 1/343 = 1/343 × 1/343 =
Answer
First read each number off the power line as a power of 7, then just add or subtract the exponents.
Power
7–4
7–3
7–2
7–1
70
71
72
73
74
75
76
77
Value
1/2401
1/343
1/49
1/7
1
7
49
343
2401
16807
117649
823543
Question
In powers of 7
Answer
2,401 × 49
74 × 72 = 74+2
76 = 1,17,649
493
(72)3 = 72×3
76 = 1,17,649
343 × 2,401
73 × 74 = 73+4
77 = 8,23,543
16,807 ÷ 49
75 ÷ 72 = 75–2
73 = 343
7 ÷ 343
71 ÷ 73 = 71–3
7–2 = 1/49
16,807 ÷ 8,23,543
75 ÷ 77 = 75–7
7–2 = 1/49
1,17,649 × 1/343
76 × 7–3 = 76–3
73 = 343
1/343 × 1/343
7–3 × 7–3 = 7–3–3
7–6 = 1/1,17,649
Why it happens: notice how much arithmetic vanished. 343 × 2,401 is an awkward multiplication of six-figure size, but as 73 × 74 it is the sum 3 + 4. Turning multiplication into addition by working with exponents is the idea behind logarithms, slide rules and the whole language of scientific notation later in this chapter.
Check it yourself: the fourth and sixth rows give the same answer, 7–2. That is not a coincidence — in both, the numerator is two steps below the denominator on the power line.