CHAPTER 2 - Power Play
nᵃ is n × n × n × n × ... × n (n multiplied by itself a times) and n⁻ᵃ = 1na .
nᵃ × nᵇ = nᵃ⁺ᵇ
(nᵃ)ᵇ = (nᵇ)ᵃ = nᵃ×ᵇ
nᵃ ÷ nᵇ = nᵃ⁻ᵇ (n ≠ 0)
nᵃ × mᵃ = (n × m)ᵃ
nᵃ ÷ mᵃ = (n ÷ m)ᵃ (m ≠ 0)
n⁰ = 1 (n ≠ 0)
The exponents could be negative also and we can convert them to positive by the following method.This shows that for any non-zero negative integers a,
where m is the positive integer and am is the multiplicative inverse of a-m.
The scientific notation for the number 308100000 is 3.081 × 10⁸. The standard form of the scientific notation of any number is x × 10ʸ, where x ≥ 1 and x < 10, and y is an integer.
n × n = n² (read as ‘n squared’ or ‘n _______________’)
na to denote n multiplied by itself ____ times
a × a × a × b × b can be expressed as _______
a) a²b³ b) ab c) a²b² d) a³b²Which 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. (a) 10v (b) 210 (c) 210v (d) 10²v
Express the number 32400 as a product of its prime factors and represent the prime factors in their exponential form
What is (-1)5 ? Is it positive or negative? What about (-1)56 ?
Is (-2)4 = 16? Verify
1.Express the following in exponential form: (i) 6 × 6 × 6 × 6 (ii) (iii) b × b × b × b y × y (iv) 5 × 5 × 7 × 7 × 7 (v) 2 × 2 × a × a (vi) a × a × a × c × c × c × c × d
2. Express each of the following as a product of powers of their prime factors in exponential form. (i) 648 (ii) 405 (iii) 540 (iv) 3600
3. Write the numerical value of each of the following: (i) 2 × 10³ (ii) (iv) (– 3)² × (– 5)² (ii) 7² × 2³ (v) 3² × 104 (iii) 3 × 44 (vi) (-2)5× (-10)6
37 can also be written as 3² × 35
Write the product p4 × p6 in exponential form.
Use this observation to compute the following. (i) 29 (ii) 57 (iii) 46
74 =?
Is 210 also equal to (25)²? Write it as a product.
Write the following expressions as a power of a power in at least two different ways: (i) 86 (ii) 715 (iii) 914 (iv) 58
In the middle of a beautiful, magical pond lies a bright pink lotus. The number of lotuses doubles every day in this pond. After 30 days, the pond is completely covered with lotuses. On which day was the pond half full? If the pond is completely covered by lotuses on the 30th day, how much of it is covered by lotuses on the 29th day? Since the number of lotuses doubles every day, the pond should be half covered on the 29th day.Write the number of lotuses (in exponential form) when the pond was — (i) fully covered (ii) half covered.
There is another pond in which the number of lotuses triples every day. When both the ponds had no flowers, Damayanti placed a lotus in the doubling pond. After 4 days, she took all the lotuses from there and put them in the tripling pond. How many lotuses will be in the tripling pond after 4 more days?What if Damayanti had changed the order in which she placed the flowers in the lakes? How many lotuses would be there?
Simplify 10454 and write it in exponential form.
Estu has 4 dresses and 3 caps. How many different ways can Estu combine the dresses and caps? Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. How many different ways can Roxie dress up?
Estu and Roxie came across a safe containing old stamps and coins that their great-grandfather had collected. It was secured with a 5-digit password. Since nobody knew the password, they had no option except to try every password until it opened. They were unlucky and the lock only opened with the last password, after they had tried all possible combinations. How many passwords did they end up checking?How many 5-digit passwords are possible? How many passwords are possible with such a lock?
What is 2100 ÷ 225 in powers of 2?
what is 20?
Write equivalent forms of the following. (i) 2-4(ii) 10-5 (iv) (-5)-3 (v) 10-100 (iii) (-7)-2
Simplify and write the answers in exponential form. (i) 2-4 × 27 (ii) 32× 3-5× 36 (iv) 24 × (-4)-2 (v) 8p × 8q
Can we say that 16384 (4^7) is 16 (4²) times larger than 1,024 (4^5)?
How many times larger than 4^–2 is 4^2?
Use the power line for 7 to answer the following questions
arrange the powers of 4 along a line
Write these numbers in the same way: (i) 172, (ii) 5642, (iii) 6374 (iv) 47561
Write using powers of 10 (i) 561.903?
Write in scientific notation (i) The Sun is located 30,00,00,00,00,00,00,00,00,000 m from the centre of our Milky Way galaxy. (ii) The number of stars in our galaxy is 1,00,00,00,00,000. (iii) The mass of the Earth is 59,76,00,00,00,00,00,00,00,00,00,000 kg. (iv) 5900 (v) 20800 (vi) 80,00,000
The distance between the Sun and Saturn is 14,33,50,00,00,000 m = 1.4335 × 1012 m. The distance between Saturn and Uranus is 14,39,00,00,00,000 m = 1.439 × 1012 m. The distance between the Sun and Earth is 1,49,60,00,00,000 m = 1.496 × 1011 m. Can you say which of the three distances is the smallest?
In scientific notation or scientific form (also called standard form),we write numbers as x × 10y, where x ≥ 1 and x < 10 is the coefficient and y, the exponent, is any integer. Often, the exponent y is more important than the coefficient x
The number line below shows the distance between the Sun and Saturn (1.4335 × 1012 m). On the number line below, mark the relative position of the Earth. The distance between the Sun and the Earth is 1.496 ×1011 m.
Express the following numbers in standard form. (i) 59,853 (ii) (iii) 34,30,000 65,950 (iv) 70,04,00,00,000
How many times can a person circum navigate (go around the world) the Earth in their lifetime if they walk non-stop? Consider the distance around the Earth as 40,000 km
Calculate and write the answer using scientific notation: (i) If one star is counted every second, how long would it take to count all the stars in the universe? Answer in terms of the number of seconds using scientific notation. (ii) If one could drink a glass of water (200 ml) every 10 seconds, how long would it take to finish the entire volume of water on Earth?
Find out the units digit in the value of 2224 ÷ 432? [Hint: 4 = 2²]
There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would be there after 40 days?
Write the given number as the product of two or more powers in three different ways. The powers can be any integers. (i) 643 (ii) 1928 (iii) 32-5
Examine each statement below and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning. (i) Cube numbers are also square numbers. (ii) Fourth powers are also square numbers. (iii) The fifth power of a number is divisible by the cube of that number. (iv) The product of two cube numbers is a cube number. (v) q46 is both a 4th power and a 6th power (q is a prime number)
Examine each statement below and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning. (i) Cube numbers are also square numbers. (ii) Fourth powers are also square numbers. (iii) The fifth power of a number is divisible by the cube of that number. (iv) The product of two cube numbers is a cube number. (v) q46 is both a 4th power and a 6th power (q is a prime number)
Simplify and write these in the exponential form. (i) 10-2 × 10-5 (ii) 57 ÷ 54 (iii) 9-7 ÷ 94 (iv) (13-2)-3 (v) m5n12 (mn)9
If 12² = 144 what is (i) (1.2)² (iii) (0.012)² (ii) (0.12)² (iv) 120²
Circle the numbers that are the same — (i) 24 × 36 (ii) 64 × 3² (iii) 610 (iv) 182 × 62 (v) 624
Identify the greater number in each of the following — (i) 4² or 34
(ii) 28 or 82 (iii) 1002 or 2100
A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits 0–9, how many digits should the code consist of?
64 is a square number (8²) and a cube number (4³). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?
A digital locker has an alphanumeric (it can have both digits and letters) passcode of length 5. Some example codes are G89P0, 38098, BRJKW, and 003AZ. How many such codes are possible?
The worldwide population of sheep (2024) is about 109, and that of goats is also about the same. What is the total population of sheep and goats? (ii) 109(iv) 1018 (ii) 1011 (v) 2 × 109 (iii) 1010 (vi) 109+ 109
Calculate and write the answer in scientific notation: (i) If each person in the world had 30 pieces of clothing, find the total number of pieces of clothing. (ii) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees. (iii) The human body has about 38 trillion bacterial cells. Find the bacterial population residing in all humans in the world. (iv) Total time spent eating in a lifetime in seconds.
What was the date 1 arab/1 billion seconds ago?
Assertion and Reasoning
(A) Both true and R explains A
(B) Both true but R does not explain A
(C) A true but R false
(D) A false but R true
A: Doubling every day is an example of exponential growth.
R: In exponential growth, the quantity increases by a fixed multiple each time.A: 2¹⁰ = 1024 means 2 multiplied by itself 10 times equals 1024.
R: The base tells how many times to multiply the exponent by itself.A: Scientific notation expresses numbers as x × 10ⁿ with 1 ≤ x < 10.
R: This makes it easier to write and compare very large or small numbers.A: In the magical pond, the pond is one-quarter full on Day 28.
R: Each day’s lotus count is double the previous day’s.A: Linear growth can overtake exponential growth if given enough time.
R: In exponential growth, increase per step decreases over time.A: aᵐ × aⁿ = aᵐ⁺ⁿ for all integers m and n.
R: Multiplying powers with the same base adds the exponents.A: a⁻ⁿ = 1/aⁿ for a ≠ 0.
R: Negative exponents represent reciprocals.A: The number 64 is both a perfect square and a perfect cube.
R: 64 = 8² and 64 = 4³.A: The cube of any even number is odd.
R: Cube of even = even × even × even = even.A: The largest 3-digit power of 2 is 2⁹.
R: 2⁹ = 512 and 2¹⁰ = 1024.A: Multiplying by 10³ increases a number’s value by 1000.
R: 10³ = 1000, so it multiplies the number by 1000.A: (aᵐ)ⁿ = aᵐⁿ holds for all integers m, n.
R: Raising a power to another power multiplies exponents.A: The number of 6-letter passwords from A–Z is 26⁶.
R: Each letter has 26 choices, independent of others.A: Linear growth means adding the same amount each step.
R: Linear growth is faster than exponential growth for small step sizes.A: In scientific notation, the exponent tells the number of decimal shifts.
R: Shifting decimal right means negative exponent.
Case-Based Study
Case Study 1 —: The Magical Folding Paper / Paper Folding to the Moon
A sheet of paper has a thickness of 0.001 cm. When you fold it once, its thickness doubles. This doubling continues with every fold.
After 10 folds, the thickness is 1.024 cm (just above 1 cm).
After 17 folds, it reaches about 131 cm (~4 feet).
After 26 folds, it becomes about 670 m.
After 30 folds, the thickness is about 10.7 km (the height at which planes fly).
After 46 folds, the paper would astonishingly reach a thickness of more than 7,00,000 km, which is nearly the distance from Earth to the Moon!
This phenomenon illustrates exponential growth, where the thickness doubles with each fold.
Q1. The thickness of a sheet after 1 fold is:
(a) 0.001 cm (b) 0.002 cm (c) 0.01 cm (d) 0.1 cm
Q2. After 10 folds, the thickness of the paper becomes about:
(a) 1 cm (b) 10 cm (c) 100 cm (d) 0.1 cm
Q3. After 17 folds, the thickness is approximately 131 cm. This is closest to:
(a) the length of a pencil (b) the height of a chair (c) the height of a person (d) the height of a door
Q4. After 26 folds, the paper’s thickness is about 670 m. Which of the following is closest to this height?
(a) Eiffel Tower (330 m) (b) Burj Khalifa (830 m) (c) Qutub Minar (73 m) (d) Empire State Building (443 m)
Q5. After 30 folds, the thickness is about 10.7 km. This is comparable to:
(a) The Mariana Trench depth (b) The height at which airplanes fly (c) The height of Mount Everest
(d) The depth of a swimming pool
Q6. If the initial thickness is 0.001 cm, then after 20 folds it will be approximately:
(a) 1 m (b) 10 m (c) 100 m (d) 1 km
Q7. The thickness of the sheet increases by how many times after every 3 folds?
(a) 2 times (b) 4 times (c) 6 times (d) 8 times
Q8. Exponential growth means:
(a) The thickness increases by a fixed number each time. (b) The thickness doubles after every fold.
(c) The thickness decreases each fold. (d) The thickness remains constant.
Q9. After 46 folds, the paper’s thickness is about 7,00,000 km. This distance is enough to reach:
(a) The Sun (b) The Moon (c) The Mariana Trench (d) Mount Everest
Q10. The growth pattern of the paper’s thickness follows which mathematical rule?
(a) Arithmetic progression (b) Linear growth (c) Geometric progression (d) Subtraction pattern
Q11: Write the thickness after 10 folds in cm using exponents.
Q12: Find the thickness after 46 folds in km.
Q13: Compare exponential growth in folding to linear growth of stacking 46 sheets without folding.
Case-Study 2: The King’s Diamonds
A king has 3 daughters. Each daughter has 3 baskets. Each basket has 3 keys. Each key opens 3 rooms. In each room, there are 3 diamonds. To find the total number of diamonds, we multiply repeatedly:
3×3=9 9×3=27 27×3=81 81×3=243
Thus, the total number of diamonds is: 37=3×3×3×3×3×3×3=2187
This shows how repeated multiplication leads to powers of 3.
Q1. The King has 3 daughters. Each daughter has 3 baskets. How many baskets are there in total?
(a) 3 (b) 6 (c) 9 (d) 27
Q2. Each basket has 3 keys. The total number of keys is:
(a) 9 (b) 18 (c) 27 (d) 81
Q3. Each key opens 3 rooms. The total number of rooms is:
(a) 27 (b) 34 (c) 81 (d) 243
Q4. Each room contains 3 diamonds. The total number of diamonds is:
(a) 243 (b) 729 (c) 2187 (d) 6561
Q5. The total number of diamonds can be expressed as:
(a) 34 (b) 35 (c) 36 (d) 37
Q6. If we had computed up to 34=81, then to find 37, we multiply 81 by:
(a) 3 (b) 9 (c) 27 (d) 243
Q7. Which mathematical concept is being illustrated in this problem?
(a) Arithmetic progression (b) Powers & exponents (c) Fractions (d) Decimals
Case Study 3: The Magical Lotus Pond
In the middle of a magical pond lies a bright pink lotus. The number of lotuses doubles every day. After 30 days, the pond is completely covered with lotuses.
On which day was the pond half full?
If the pond is completely covered by lotuses on the 30th day, how much of it is covered on the 29th day?
Since the number of lotuses doubles every day, the pond should be half covered on the 29th day.
Lotuses in exponential form:
Fully covered → 2³⁰ Half covered → 2²⁹
There is another pond where the number of lotuses triples every day. When both ponds had no flowers, Damayanti placed a lotus in the doubling pond. After 4 days, she took all the lotuses from there and placed them in the tripling pond. After 4 more days, the number of lotuses becomes:
1 × 2⁴ × 3⁴ = 16 × 81 = 1296
If Damayanti had started with the tripling pond first and then moved to the doubling pond, the result would be:
1 × 3⁴ × 2⁴ = 81 × 16 = 1296
Thus, the order does not change the final number.
Q1. If the pond is completely covered on the 30th day, on which day was it half covered?
(a) 28th day (b) 29th day (c) 30th day (d) 31st day
Q2. The pond was fully covered with 2³⁰ lotuses. How many lotuses were there on the 29th day?
(a) 2²⁸ (b) 2²⁹ (c) 2³⁰ (d) 2³¹
Q3. On the 30th day, the pond had 2³⁰ lotuses. On the 29th day, it had:
(a) (2³⁰ ÷ 2) (b) 2 × 2³⁰ (c) 2³⁰ + 2 (d) (2³⁰ ÷ 4)
Q4. After 4 days in the doubling pond, Damayanti had:
(a) 2² = 4 lotuses (b) 2³ = 8 lotuses (c) 2⁴ = 16 lotuses (d) 2⁵ = 32 lotuses
Q5. After moving them to the tripling pond for 4 more days, the total becomes:
(a) 16 × 3² (b) 16 × 3³ (c) 16 × 3⁴ (d) 16 × 3⁵
Q6. The final number of lotuses after both transfers is:
(a) 648 (b) 1296 (c) 2187 (d) 4096
Q7. If Damayanti had started with the tripling pond first and then moved to the doubling pond, the result would be:
(a) Different (b) Same (c) Half (d) Double
Magical Pond & Doubling
A magical pond has 1 lotus flower on Day 1. Each day, the number of lotus flowers doubles. On Day 30, the pond is completely covered.
Q1.1 On which day is the pond half-covered?
Q1.2 Write the number of lotuses on Day 30 in exponential form.
Q1.3 If each lotus takes up 0.5 m², what is the total area covered by lotuses on Day 30?
Case-Study 4: Dresses, Caps & Shoes
Estu has 4 dresses and 3 caps For each cap, he can choose any of the 4 dresses.So, the total number of combinations is: 4 × 3 = 12 We can also see this as: For each dress, he can choose any of the 3 caps. So, 3 × 4 = 12 Both ways give the same answer.Now, Roxie has 7 dresses, 2 hats, and 3 pairs of shoes.For each dress, she can choose 1 of 2 hats and 1 of 3 pairs of shoes.The total number of ways is: 7 × 2 × 3 = 42
Q1. Estu has 4 dresses and 3 caps. How many different outfits can he make?
(a) 7 (b) 12 (c) 16 (d) 24
Q2. Which expression represents Estu’s combinations?
(a) 4 + 3 (b) 4 × 3 (c) 4³ (d) 3⁴
Q3. Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. How many different outfits can she make?
(a) 12 (b) 24 (c) 42 (d) 84
Q4. Which expression correctly shows the number of Roxie’s combinations?
(a) 7 + 2 + 3 (b) 7 × 2 × 3 (c) 7² × 3 (d) 7 × 2³
Q5. If Estu had 5 dresses and 4 caps instead, the total combinations would be:
(a) 5 × 4 = 20 (b) 5 + 4 = 9 (c) 5⁴ (d) 4⁵
Q6. The concept used in both Estu’s and Roxie’s problems is:
(a) Addition principle (b) Multiplication principle (c) Division principle (d) Subtraction principle
Case Study 5: Password
Estu and Roxie discovered a safe secured with a 5-digit password. Since nobody knew the password, they had to try every possibility until it opened. Unfortunately, it only opened with the last possible password.To understand this, they considered simpler locks: A 2-digit lock has 10×10=10²=100 passwords. A 3-digit lock has 10×10×10=10³=1000 passwords. Similarly, a 5-digit lock has 105 passwords.Later, Estu thought of a 6-letter lock using A–Z (26 letters).
Q1. How many passwords are possible in a 2-digit lock?
(a) 10 (b) 20 (c) 10² =100 (d) 200
Q2. How many passwords are possible in a 3-digit lock?
(a) 10²=100 (b) 10³=1000 (c) 999 (d) 900
Q3. For the 5-digit lock, how many passwords did Estu and Roxie check before it opened?
(a) 9999 (b) 50000 (c) 105=100000 (d) 510
Q4. If Estu buys a lock with 6 slots using A–Z, how many passwords are possible?
(a) 66 (b) 266 (c) 106 (d) 26 × 6
Q5. Which of the following is an example of a combinatorial password/coding system?
(i) PIN codes in India (ii) Mobile numbers (iii) Vehicle registration numbers
Choose the correct option:
(a) Only (i) (b) Only (ii) (c) (i) and (ii) (d) (i), (ii), and (iii)
Q6: How many different codes are possible?
Q7: If the lock instead uses 5 characters (A–Z letters only), how many codes are possible?
Q8: Which is more secure and why?
Case Study 6 — Roxie’s Tulābhāra Donation
Nanjundappa wants to donate jaggery equal to Roxie’s weight and wheat equal to Estu’s weight. Worth of jaggery (₹) = Roxie’s weight (kg) × cost of 1 kg jaggery . Worth of wheat (₹) = Estu’s weight (kg) × cost of 1 kg wheat. Assumptions: Roxie’s weight = 45 kg, cost of 1 kg jaggery = ₹70 . Estu’s weight = 50 kg, cost of 1 kg wheat = ₹50 . Thus, Worth of jaggery = 45 × 70 = ₹3150. Worth of wheat = 50 × 50 = ₹2500 This practice of offering goods equal to one’s weight is called Tulābhāra.Roxie wonders: “Instead of jaggery, if we use 1-rupee coins, how many coins are needed to equal my weight?”
Q1. What is the worth of jaggery donated by Nanjundappa?
(a) ₹2500 (b) ₹3000 (c) ₹3150 (d) ₹3500
Q2. What is the worth of wheat donated by Nanjundappa?
(a) ₹2000 (b) ₹2500 (c) ₹2700 (d) ₹3000
Q3. If Roxie’s weight is 45 kg and each ₹1 coin weighs 5 g, how many coins are required to equal her weight?
(a) 450 coins (b) 4500 coins (c) 9000 coins (d) 45 × 200 = 9000 coins
Q4. Tulābhāra is a practice symbolizing —
(a) Wealth accumulation (b) Bhakti (surrendering and gratitude) (c) Business transaction (d) Entertainment
Q5. If Estu’s weight is 50 kg and instead of wheat, 2-rupee coins are used, assuming each coin weighs 5 g, how many coins are required?
(a) 5000 coins (b) 10000 coins (c) 50 × 1000 ÷ 2 = 25000 coins (d) 50 × 200 = 10000 coins
Q6: Find the worth of the donation.
Q7: If 1-rupee coins replace the jaggery (mass = 4 g each), find the number of coins needed.
Q8: Write your answer in scientific notation.
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