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

Book page 242 Updated on2026-09-05

Q1.
3⁷ can also be written as 3² × 3⁵. Can you reason out why?
Answer

Because seven 3s can be split into two 3s and five 3s, and it makes no difference where you put the brackets.

37 = 3 × 3 × 3 × 3 × 3 × 3 × 3
= (3 × 3) × (3 × 3 × 3 × 3 × 3)
= 32 × 35
Check: 9 × 243 = 2187 = 37 ✓

Every split of 7 into two counting numbers works equally well: 31 × 36, 33 × 34, 34 × 33. All of them give 2187.

Why it happens: multiplication is associative — regrouping the factors never changes the product. So a string of a + b equal factors can always be cut into a group of a and a group of b. That single fact is the whole content of na × nb = na+b: the exponents add because the counts of factors add.
Q2.
Write the product p⁴ × p⁶ in exponential form.
Answer
p4 × p6 = (p × p × p × p) × (p × p × p × p × p × p)
= p multiplied by itself 4 + 6 = 10 times
= p10
Why it happens: nothing about the argument used the fact that the base was a number. p is a letter-number standing for any value, and the count of factors is still 4 + 6. This is how the chapter passes from particular cases such as 34 × 33 to the general rule na × nb = na+b, where a and b are counting numbers.
Tip: the rule needs the same base. p4 × q6 cannot be shortened at all; only p4 × p6 can.
Q3.
Use this observation to compute the following. (i) 2⁹ (ii) 5⁷ (iii) 4⁶
Answer

Split each exponent into two parts you already know, then multiply once.

(i) 29 = 24 × 25 = 16 × 32 = 512
(also 23 × 26 = 8 × 64 = 512)
(ii) 57 = 53 × 54 = 125 × 625 = 78125
(also 52 × 55 = 25 × 3125 = 78125)
(iii) 46 = 43 × 43 = 64 × 64 = 4096
(also 42 × 44 = 16 × 256 = 4096)
Why it happens: the rule turns nine multiplications into one. You only need to remember a few small powers — 24, 25, 53, 54, 43 — and every larger power of the same base is within one multiplication of them. Since the split is your choice, pick the two pieces you know best.
Q4.
Is 2¹⁰ also equal to (2⁵)² ? Write it as a product.
Answer

Yes.

210 = (2 × 2 × 2 × 2 × 2) × (2 × 2 × 2 × 2 × 2)
= (25) × (25)
= (25)2
Check: 32 × 32 = 1024 = 210 ✓

The same ten 2s can also be grouped in fives of two: 210 = (22)5 = 4 × 4 × 4 × 4 × 4 = 1024. Both groupings are legal, so (25)2 = (22)5 = 210.

Why it happens: (na)b says "take b copies of a block of a factors". Altogether that is a × b factors — and a × b does not care about the order of a and b. That is why (na)b = (nb)a = nab. Note the contrast with the product rule: multiplying powers adds exponents, raising a power to a power multiplies them.
Q5.
Write the following expressions as a power of a power in at least two different ways: (i) 8⁶ (ii) 7¹⁵ (iii) 9¹⁴ (iv) 5⁸
Answer

Split the exponent into two factors (that keeps the base), or rewrite the base as a power (that changes the base).

Same base — split the exponentNew base — split the base
(i) 86(82)3 = (83)28 = 23, so 86 = (23)6 = 218 = (29)2
(ii) 715(73)5 = (75)315 = 15 × 1, so also (715)1
(iii) 914(92)7 = (97)29 = 32, so 914 = (32)14 = 328 = (34)7 = (37)4
(iv) 58(52)4 = (54)2 = ((52)2)25 is prime, so the base cannot be split
Why it happens: the number of ways you can write nm as a power of a power is exactly the number of ways of factorising m — 15 = 3 × 5 = 5 × 3 gives two, while a prime exponent gives essentially none. If the base is itself a power, as with 8 = 23 and 9 = 32, you get extra choices, because (23)6 = 218 reopens the whole game with the new exponent 18.
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