Class 8 Maths Power Play Notes

Chapter 2: Power Play (Maths)

Overview:
This chapter focuses on powers and exponents. It teaches how to express large numbers in a simpler form using powers, understand their properties, and apply them in calculations.

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Key Concepts:

1. Powers and Exponents:
   • A power (or exponent) tells us how many times a number (called the base) is multiplied by itself.
     • Example: $2^3 = 2 \times 2 \times 2 = 8$23=2×2×2=8
       • Here, 2 is the base and 3 is the exponent.
2. Laws of Exponents:
   Exponents follow special rules that make calculations easier:
   • Product of powers: $a^m \times a^n = a^{m+n}$am×an=am+n
   • Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$anam​=am−n (a ≠ 0)
   • Power of a power: $(a^m)^n = a^{m \times n}$(am)n=am×n
   • Power of a product: $(ab)^m = a^m \times b^m$(ab)m=am×bm
   • Power of a quotient: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$(ba​)m=bmam​ (b ≠ 0)
3. Zero and Negative Exponents:
   • Any non-zero number raised to the power 0 is 1: $a^0 = 1$a0=1
   • Negative exponents mean reciprocal: $a^{-n} = \frac{1}{a^n}$a−n=an1​
4. Expressing Large Numbers Using Powers of 10:
   • Large and small numbers can be expressed in standard form using powers of 10.
     • Example: 5000 = $5 \times 10^3$5×103
5. Applications:
   • Powers simplify calculations with very large or very small numbers.
   • Useful in scientific notation, algebra, and real-life problems like population or distance calculations.

Questions

A. Very Short Answer Questions (1–10)

1. Write $2 \times 2 \times 2 \times 2$2×2×2×2 in exponential form.
2. Find the value of $3^4$34.
3. Write $5^0$50 as a number.
4. Express $1000$1000 as a power of 10.
5. Simplify $7^1$71.
6. Find the value of $2^3 \times 2^2$23×22.
7. Simplify $\frac{5^4}{5^2}$5254​.
8. Write $0.01$0.01 in standard form using powers of 10.
9. Find the value of $(-3)^2$(−3)2.
10. Express $1$1 as a power of any non-zero number.

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B. Short Answer Questions (11–25)

11. Simplify $(2^3)^2$(23)2.
12. Simplify $(5^2)^3$(52)3.
13. Calculate $(3 \times 2)^4$(3×2)4 using the law of exponents.
14. Simplify $\frac{10^5}{10^3}$103105​.
15. Express 0.0005 in standard form.
16. Simplify $(x^3 \times x^2)$(x3×x2).
17. Simplify $\frac{a^7}{a^4}$a4a7​.
18. Express $\frac{1}{8}$81​ as a power of 2.
19. Simplify $(2^3)^0$(23)0.
20. Evaluate $4^{-1}$4−1.
21. Simplify $(x^0 + y^0)$(x0+y0).
22. Express 4500 in standard form.
23. Simplify $\frac{2^5 \times 3^2}{2^2}$2225×32​.
24. Write $(7^3)^2$(73)2 as a single power.
25. Find the value of $(5^0 + 6^0)$(50+60).

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C. Long Answer / Word Problems (26–40)

26. A bacterium doubles in number every hour. If it starts with 1 bacterium, write the number of bacteria after 5 hours using powers of 2.
27. A population of fish is $3^4$34. If it triples every year, what will it be after 2 years?
28. A cube has side length 2 cm. Write the formula for volume using powers and calculate it.
29. Express $0.00009$0.00009 in standard form.
30. Simplify $(2^3 \times 5^3)$(23×53).
31. The distance to a star is $3 \times 10^6$3×106 km. Express it in standard form.
32. Simplify $\frac{(x^5 y^3)^2}{x^3 y}$x3y(x5y3)2​.
33. If $a = 2$a=2 and $b = 3$b=3, evaluate $a^2 b^3$a2b3.
34. Simplify $(2^3 \times 3^3)^2$(23×33)2.
35. Write 0.000007 in standard form.
36. The mass of a grain of rice is $10^{-3}$10−3 kg. Express in grams.
37. Calculate $(3^2 \times 4^2) \div (6^2)$(32×42)÷(62).
38. A virus multiplies 10 times every day. If it starts with 1, write its population after 3 days using powers of 10.
39. Express $\frac{1}{1000}$10001​ as a negative power of 10.
40. Simplify $(2^2 \times 5)^3$(22×5)3.

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D. Higher Order / Thinking Questions (41–50)

41. Simplify $(x^2 y^3)^0$(x2y3)0.
42. Compare $2^5$25 and $5^2$52. Which is larger?
43. Find the missing exponent: $7^? = 343$7?=343.
44. Simplify $(3x^2)^3 \times (2x^3)^2$(3×2)3×(2×3)2.
45. Express $0.00056$0.00056 in standard form and round to 2 significant figures.
46. If $a^m \times a^n = a^{12}$am×an=a12 and $m = 7$m=7, find $n$n.
47. Simplify $(2^4)^3 \div 2^5$(24)3÷25.
48. Write a number greater than 1 but less than 10 in standard form using powers of 10.
49. Evaluate $(10^3)^2 \div 10^4$(103)2÷104.
50. If $x^{-2} = 25$x−2=25, find the value of x.

Answers for Chapter 2: Power Play – Class 8 (50 Questions)

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A. Very Short Answer Questions (1–10)

1. $2^4$24
2. $3^4 = 3 \times 3 \times 3 \times 3 = 81$34=3×3×3×3=81
3. $5^0 = 1$50=1
4. $1000 = 10^3$1000=103
5. $7^1 = 7$71=7
6. $2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$23×22=23+2=25=32
7. $\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25$5254​=54−2=52=25
8. $0.01 = 1 \times 10^{-2}$0.01=1×10−2
9. $(-3)^2 = (-3) \times (-3) = 9$(−3)2=(−3)×(−3)=9
10. $1 = a^0$1=a0 (for any $a \neq 0$a=0)

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B. Short Answer Questions (11–25)

11. $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$(23)2=23×2=26=64
12. $(5^2)^3 = 5^{2 \times 3} = 5^6 = 15625$(52)3=52×3=56=15625
13. $(3 \times 2)^4 = 6^4 = 1296$(3×2)4=64=1296
14. $\frac{10^5}{10^3} = 10^{5-3} = 10^2 = 100$103105​=105−3=102=100
15. $0.0005 = 5 \times 10^{-4}$0.0005=5×10−4
16. $x^3 \times x^2 = x^{3+2} = x^5$x3×x2=x3+2=x5
17. $\frac{a^7}{a^4} = a^{7-4} = a^3$a4a7​=a7−4=a3
18. $\frac{1}{8} = 2^{-3}$81​=2−3
19. $(2^3)^0 = 2^{3 \cdot 0} = 2^0 = 1$(23)0=23⋅0=20=1
20. $4^{-1} = \frac{1}{4} = 0.25$4−1=41​=0.25
21. $x^0 + y^0 = 1 + 1 = 2$x0+y0=1+1=2
22. $4500 = 4.5 \times 10^3$4500=4.5×103
23. $\frac{2^5 \times 3^2}{2^2} = 2^{5-2} \times 3^2 = 2^3 \times 3^2 = 8 \times 9 = 72$2225×32​=25−2×32=23×32=8×9=72
24. $(7^3)^2 = 7^{3 \cdot 2} = 7^6 = 117649$(73)2=73⋅2=76=117649
25. $5^0 + 6^0 = 1 + 1 = 2$50+60=1+1=2

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C. Long Answer / Word Problems (26–40)

26. Number of bacteria after 5 hours: $2^5 = 32$25=32
27. Population after 2 years: Initial = $3^4 = 81$34=81, triples each year: $81 \times 3 \times 3 = 81 \times 9 = 729$81×3×3=81×9=729
28. Cube side = 2 cm, Volume = $2^3 = 8 \text{ cm}^3$23=8 cm3
29. $0.00009 = 9 \times 10^{-5}$0.00009=9×10−5
30. $2^3 \times 5^3 = (2 \times 5)^3 = 10^3 = 1000$23×53=(2×5)3=103=1000
31. Distance $3 \times 10^6$3×106 km = already in standard form.
32. $\frac{(x^5 y^3)^2}{x^3 y} = \frac{x^{10} y^6}{x^3 y} = x^{10-3} y^{6-1} = x^7 y^5$x3y(x5y3)2​=x3yx10y6​=x10−3y6−1=x7y5
33. $a^2 b^3 = 2^2 \cdot 3^3 = 4 \cdot 27 = 108$a2b3=22⋅33=4⋅27=108
34. $(2^3 \times 3^3)^2 = (2 \cdot 3)^6 = 6^6 = 46656$(23×33)2=(2⋅3)6=66=46656
35. $0.000007 = 7 \times 10^{-6}$0.000007=7×10−6
36. Mass = $10^{-3}$10−3 kg = $10^{-3} \times 1000$10−3×1000 g = 1 g
37. $(3^2 \times 4^2) \div 6^2 = (9 \times 16)/36 = 144/36 = 4$(32×42)÷62=(9×16)/36=144/36=4
38. Virus population: Day 3 = $1 \times 10^3 = 1000$1×103=1000
39. $\frac{1}{1000} = 10^{-3}$10001​=10−3
40. $(2^2 \times 5)^3 = (4 \cdot 5)^3 = 20^3 = 8000$(22×5)3=(4⋅5)3=203=8000

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D. Higher Order / Thinking Questions (41–50)

41. $(x^2 y^3)^0 = 1$(x2y3)0=1
42. $2^5 = 32$25=32, $5^2 = 25$52=25 → $2^5 > 5^2$25>52
43. $7^? = 343$7?=343 → $7^3 = 343$73=343, so exponent = 3
44. $(3x^2)^3 \times (2x^3)^2 = 27 x^6 \cdot 4 x^6 = 108 x^{12}$(3×2)3×(2×3)2=27×6⋅4×6=108×12
45. $0.00056 = 5.6 \times 10^{-4}$0.00056=5.6×10−4 → 2 significant figures = $5.6 \times 10^{-4}$5.6×10−4
46. $a^m \cdot a^n = a^{12}$am⋅an=a12, $m = 7$m=7 → $7+n=12 \Rightarrow n = 5$7+n=12⇒n=5
47. $(2^4)^3 \div 2^5 = 2^{12}/2^5 = 2^{12-5} = 2^7 = 128$(24)3÷25=212/25=212−5=27=128
48. Example: 7 → $7 \times 10^0$7×100
49. $(10^3)^2 \div 10^4 = 10^6 / 10^4 = 10^{6-4} = 10^2 = 100$(103)2÷104=106/104=106−4=102=100
50. $x^{-2} = 25 \Rightarrow x^2 = \frac{1}{25} \Rightarrow x = \frac{1}{5}$x−2=25⇒x2=251​⇒x=51​