Chapter 2: Power Play (NCERT Ganita Prakash)
1. The thickness of a sheet of paper after 10 folds, starting from thickness \(v\) cm, is expressed as
(a) \(10v\) (b) \(10 + v\) (c) \(2 \times 10 \times v\) (d) \(2^{10}v\)
2. Which of the following is equal to \(2^0\)?
(a) 0 (b) 1 (c) 2 (d) undefined
3. \(3^4 \times 3^5\) is equal to
(a) \(3^9\) (b) \(3^{20}\) (c) \(9^9\) (d) \(3^{45}\)
4. The standard form of 70,04,00,00,000 is
(a) \(7.004 \times 10^{10}\) (b) \(70.04 \times 10^9\) (c) \(7.004 \times 10^{11}\) (d) \(7004 \times 10^7\)
5. Assertion (A): \((-1)^5\) is negative.
Reason (R): A negative number raised to an odd power is negative.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
6. Assertion (A): \(2^{10} = (2^5)^2\).
Reason (R): \((n^a)^b = n^{ab}\).
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
7. The number of 5-digit passwords using digits 0–9 is
(a) \(10^4\) (b) \(10^5\) (c) \(5^{10}\) (d) \(10^6\)
8. \(10^{-3}\) is equal to
(a) \(1000\) (b) \(\frac{1}{1000}\) (c) \(0.001\) (d) both (b) and (c)
9. Which expression is in exponential form?
(a) \(2 \times 2 \times 2 \times 5 \times 5\) (b) \(2^3 \times 5^2\) (c) \(8 \times 25\) (d) \(200\)
10. After 30 folds the thickness of a 0.001 cm paper reaches approximately
(a) 1 km (b) 10 km (c) 100 km (d) 1000 km
11. Express \(32400\) as the product of powers of its prime factors.
12. Write \(2^{-4}\) in two equivalent forms.
13. Simplify: \(5^3 \times 5^{-7}\).
14. How many different ways can Roxie combine 7 dresses, 2 hats and 3 pairs of shoes?
15. Write \(561.903\) using powers of 10.
16. What is the value of \(n^0\) when \(n \neq 0\)? Give one example.
17. Express the following in exponential form and then find the value:
(i) \(2 \times 10^3\) (ii) \(3^2 \times 4^4\) (iii) \((-2)^5 \times (-10)^3\)
18. Simplify and write in exponential form:
(i) \(2^{-4} \times 2^7\) (ii) \(p^3 \times p^{-10}\) (iii) \((3^4)^2\)
19. A pond is completely covered with lotuses on the 30th day. The number of lotuses doubles every day. On which day was the pond half-covered? Write the number of lotuses on both days in exponential form.
20. Write the prime-factor form of 648 and 3600 in exponential notation.
21. Convert the following into standard form:
(i) 59 853 (ii) 34 30 000 (iii) 70 04 00 00 000
22. The thickness of a sheet of paper is 0.001 cm. It doubles after every fold.
(a) Write the thickness after 10 folds in exponential form.
(b) After how many folds does the thickness become approximately 10.7 km? Show the calculation using powers of 2.
(c) Compare this exponential growth with linear growth by stating how many folds would be needed if thickness increased by 0.001 cm each time to reach the same height.
23. Simplify and express in exponential form (show all steps):
(i) \(10^4 \div 5^4\) (ii) \(2^{-4} \times 2^7 \times 2^{-1}\) (iii) \((8^p \times 8^q) \div 8^{p+q}\)
24. A 5-digit lock uses digits 0–9.
(a) How many possible passwords exist?
(b) If each digit is replaced by a letter A–Z, how many passwords are possible?
(c) Which lock is safer and why? Relate your answer to exponents.
25. Magical Pond Case
A magical pond’s lotuses double every day. On the 30th day the pond is fully covered.
Sub-questions:
(a) On which day was the pond half-covered? (1 mark)
(b) Write the number of lotuses on the 30th day in exponential form. (1 mark)
(c) If another pond triples the lotuses every day and \(2^4\) lotuses are transferred to it, how many lotuses will be there after 4 more days? (1 mark)
(d) Which growth is faster — doubling or tripling? Justify using exponents. (1 mark)
26. Scientific Distances Case
Distance of Sun from Saturn = \(1.4335 \times 10^{12}\) m
Distance of Sun from Earth = \(1.496 \times 10^{11}\) m
Sub-questions:
(a) Which distance is smaller? (1 mark)
(b) Express both distances in ordinary form. (1 mark)
(c) How many times larger is the Saturn distance than the Earth distance? (1 mark)
(d) Write the distance of Sun from Uranus (\(1.439 \times 10^{12}\) m) and compare all three using exponents. (1 mark)
1. (d) — chapter states thickness after 10 folds = \(2^{10}v\)
2. (b) — \(x^0 = 1\) (\(x \neq 0\))
3. (a) — \(n^a \times n^b = n^{a+b}\)
4. (a) — standard form rule
5. (a) — both true, R explains A
6. (a) — both true, R explains A
7. (b) — \(10^5\) passwords
8. (d) — \(n^{-a} = 1/n^a\)
9. (b) — exponential notation
10. (b) — text states ≈10.7 km after 30 folds
11. \(32400 = 2^4 \times 3^4 \times 5^2\) (1 mark prime factors, 1 mark exponents)
12. \(2^{-4} = 1/2^4 = 1/16\) (1 mark each form)
13. \(5^{-4}\) (correct law 1 mark, simplification 1 mark)
14. \(7 \times 2 \times 3 = 42\) ways (multiplication rule 2 marks)
15. \(5 \times 10^2 + 6 \times 10^1 + 1 \times 10^0 + 9 \times 10^{-1} + 0 \times 10^{-2} + 3 \times 10^{-3}\) (correct powers 2 marks)
16. 1; example \(5^0 = 1\) (definition 1 mark, example 1 mark)
17. (i) 2000 (ii) 768 (iii) –32000 (formula 1 mark each, calculation 1 mark each)
18. (i) \(2^3\) (ii) \(p^{-7}\) (iii) \(3^8\) (law 1 mark, simplification 1 mark, final form 1 mark)
19. 29th day; \(2^{30}\) and \(2^{29}\) (reasoning 1 mark, exponential forms 2 marks)
20. \(648 = 2^3 \times 3^4\); \(3600 = 2^4 \times 3^2 \times 5^2\) (factorisation 1 mark each, exponents 1 mark)
21. (i) \(5.9853 \times 10^4\) (ii) \(3.43 \times 10^6\) (iii) \(7.004 \times 10^{10}\) (each correct form 1 mark)
22. (a) \(2^{10} \times 0.001\) (1 mark)
(b) 30 folds — \(2^{30} \times 0.001 \approx 10.7\) km (2 marks)
(c) Linear growth would require \(10.7 \times 10^5 / 0.001 = 10.7 \times 10^8\) additions (2 marks)
23. (i) \(2^4\) (ii) \(2^2\) (iii) 1 (each step with law 1 mark, final answer 1 mark)
24. (a) \(10^5\) (b) \(26^5\) (c) Letter lock safer because \(26^5 > 10^5\) (exponent comparison 2 marks)
25. (a) 29th day (1) (b) \(2^{30}\) (1) (c) \(2^4 \times 3^4 = 6^4\) (1) (d) Tripling faster (\(3^4 > 2^4\)) (1)
26. (a) Earth–Sun (1) (b) 1 433 500 000 000 m and 149 600 000 000 m (1) (c) \(\approx 9.58\) times (1) (d) Uranus largest, exponents show order (1)
All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.