Class 8 : Maths – Lesson 2. Power Play
EXPLANATION AND ANALYSIS
🌍 INTRODUCTION — THE IDEA OF POWER IN NUMBERS
🔋 In mathematics, we often come across numbers that are written again and again in multiplication form.
✍️ Writing the same number many times using multiplication is time-consuming and looks messy.
📘 To make such expressions short, neat, and meaningful, we use powers.
📌 This chapter explains:
what powers mean
how exponents work
how to simplify expressions using rules of powers
These ideas make calculations faster and form a strong base for algebra and science.
🔢 WHAT DOES “POWER” MEAN?
🧩 A power is a short way of writing repeated multiplication of the same number.
📖 When a number is multiplied by itself again and again, we use powers to represent it.
📝 Example idea:
Instead of writing
4 × 4 × 4 × 4
we write
4⁴
✨ This form is shorter and easier to understand.
🧮 PARTS OF A POWER
🔸 In the expression aⁿ, two parts are involved:
🟧 Base → a
🟩 Exponent (or power) → n
🧠 Meaning:
aⁿ means a is multiplied by itself n times.
📌 The base tells which number is being multiplied.
📌 The exponent tells how many times it is multiplied.
📐 SIMPLE EXAMPLES OF POWERS
✏️
2³ = 2 × 2 × 2
5² = 5 × 5
10⁴ = 10 × 10 × 10 × 10
🧾 Writing numbers using powers saves space and reduces chances of error.
🔢 POWERS OF 10
⚡ Powers of 10 are used frequently in mathematics and daily life.
📊 Examples:
10¹ = 10
10² = 100
10³ = 1000
10⁴ = 10000
🌍 Powers of 10 are useful in:
population data
distances
money calculations
measurements
✖️ MULTIPLICATION OF POWERS WITH SAME BASE
📌 When powers have the same base, multiplication becomes easy.
🧾 Rule
If the base is the same, add the exponents.
📐 Formula
If base = a,
aᵐ × aⁿ = aᵐ⁺ⁿ
✏️ Example
3² × 3³
= 3⁵
🧠 Same base → add powers.
➗ DIVISION OF POWERS WITH SAME BASE
📌 Division of powers also follows a simple rule.
🧾 Rule
If the base is the same, subtract the exponents.
📐 Formula
If base = a,
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
✏️ Example
7⁵ ÷ 7²
= 7³
🧠 Same base → subtract powers.
🔁 POWER OF A POWER
📌 Sometimes a power itself is raised to another power.
🧾 Rule
When a power is raised to another power, multiply the exponents.
📐 Formula
If base = a,
(aᵐ)ⁿ = aᵐⁿ
✏️ Example
(2³)²
= 2⁶
📌 Brackets are very important in this rule.
✖️ POWER OF A PRODUCT
📌 A product raised to a power can be simplified easily.
🧾 Rule
The power of a product is equal to the product of the powers.
📐 Formula
(ab)ⁿ = aⁿ × bⁿ
✏️ Example
(2 × 3)²
= 2² × 3²
🧠 This rule helps break complex expressions into simpler ones.
➗ POWER OF A QUOTIENT
📌 Powers can also be applied to fractions.
🧾 Rule
The power of a fraction is applied to both numerator and denominator.
📐 Formula
(a / b)ⁿ = aⁿ / bⁿ
✏️ Example
(3 / 5)²
= 3² / 5²
📌 This rule is useful while simplifying expressions involving fractions.
🔢 ZERO POWER
📌 Zero power is a special and important rule.
🧾 Rule
If a ≠ 0,
a⁰ = 1
✏️ Examples
9⁰ = 1
100⁰ = 1
🧠 Any non-zero number raised to power zero is always 1.
🔄 NEGATIVE POWERS
📌 Powers can also be negative.
🧾 Rule
a⁻ⁿ = 1 / aⁿ
✏️ Example
2⁻² = 1 / 2²
🧠 A negative power represents the reciprocal of the number.
⚠️ COMMON MISTAKES TO AVOID
🚫 Adding bases instead of exponents
🚫 Ignoring brackets in expressions
🚫 Applying rules to different bases
🚫 Forgetting rules of zero and negative powers
✔️ Always check:
the base
the exponent
the correct rule
🏠 USES OF POWERS IN DAILY LIFE
📊 Writing very large numbers
🔬 Scientific calculations
📐 Geometry and algebra
🧮 Computer and technology fields
🌍 Measurements in physics and chemistry
🌟 WHY THIS LESSON IS IMPORTANT
🏆 Makes calculations faster
🚀 Builds strong algebra foundation
🧠 Improves understanding of numbers
📘 Prepares for higher classes
🌱 Useful in real-life applications
🧾 SUMMARY
📌 Powers show repeated multiplication
📌 Base is the number being multiplied
📌 Exponent shows number of times
📌 Same base multiplication → add powers
📌 Same base division → subtract powers
📌 Zero power gives 1
📌 Negative power gives reciprocal
🔁 QUICK RECAP
🔢 Power → repeated multiplication
🟧 Base → main number
🟩 Exponent → number of times
➕ Multiply → add exponents
➖ Divide → subtract exponents
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TEXTBOOK QUESTIONS
🔒 ❓ Figure it Out
🔒 ❓ Q1. Find out the units digit in the value of 2²²⁴ ÷ 4³²? [Hint: 4 = 2²]
📌 ✅ Answer:
🟢 Step 1
⬥ 4³² = (2²)³² = 2⁶⁴, so 2²²⁴ ÷ 4³² = 2²²⁴ ÷ 2⁶⁴ = 2¹⁶⁰
🔵 Step 2
⬥ Powers of 2 have unit digits in the cycle 2, 4, 8, 6 (length 4)
⬥ 160 is divisible by 4, so the unit digit is 6
🔒 ❓ Q2. There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would be there after 40 days?
📌 ✅ Answer:
⬥ Each day adds 5 bottles
⬥ In 40 days: 40 × 5 = 200
🔒 ❓ Q3. Write the given number as the product of two or more powers in three different ways. The powers can be any integers.
🔒 ❓ (i) 64³
📌 ✅ Answer:
⬥ 64 = 2⁶, so 64³ = (2⁶)³ = 2¹⁸
⬥ Three ways as products of powers:
⬥ 64³ = 2¹⁰ × 2⁸
⬥ 64³ = 8⁴ × 2⁶
⬥ 64³ = 4⁵ × 2⁸
🔒 ❓ (ii) 192⁸
📌 ✅ Answer:
⬥ 192 = 64 × 3 = 2⁶ × 3
⬥ 192⁸ = (2⁶ × 3)⁸ = 2⁴⁸ × 3⁸
⬥ Three ways as products of powers:
⬥ 192⁸ = 2⁴⁸ × 3⁸
⬥ 192⁸ = 64⁸ × 3⁸
⬥ 192⁸ = 6⁸ × 32⁸
🔒 ❓ (iii) 32⁻⁵
📌 ✅ Answer:
⬥ 32 = 2⁵, so 32⁻⁵ = (2⁵)⁻⁵ = 2⁻²⁵
⬥ Three ways as products of powers:
⬥ 32⁻⁵ = 2⁻¹⁰ × 2⁻¹⁵
⬥ 32⁻⁵ = 4⁻⁵ × 2⁻¹⁵
⬥ 32⁻⁵ = 16⁻⁵ × 2⁻⁵
🔒 ❓ Q4. Examine each statement and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning.
🔒 ❓ (i) Cube numbers are also square numbers.
📌 ✅ Answer:
⬥ Some cube numbers are squares (e.g., 64 = 8² = 4³)
⬥ Some cube numbers are not squares (e.g., 8 = 2³)
⬥ Only Sometimes True
🔒 ❓ (ii) Fourth powers are also square numbers.
📌 ✅ Answer:
⬥ a⁴ = (a²)², which is a square for every integer a
⬥ Always True
🔒 ❓ (iii) The fifth power of a number is divisible by the cube of that number.
📌 ✅ Answer:
⬥ a⁵ ÷ a³ = a², which is an integer for nonzero a
⬥ Always True
🔒 ❓ (iv) The product of two cube numbers is a cube number.
📌 ✅ Answer:
⬥ a³ × b³ = (ab)³
⬥ Always True
🔒 ❓ (v) q⁴⁶ is both a 4th power and a 6th power (q is a prime number).
📌 ✅ Answer:
⬥ 46 is not a multiple of 4 or 6
⬥ With q prime, the exponent alone decides the power
⬥ Never True
🔒 ❓ Q5. Simplify and write these in the exponential form.
🔒 ❓ (i) 10⁻² × 10⁻⁵
📌 ✅ Answer:
⬥ 10⁻² × 10⁻⁵ = 10⁻(2+5) = 10⁻⁷
🔒 ❓ (ii) 5⁷ ÷ 5⁴
📌 ✅ Answer:
⬥ 5⁷ ÷ 5⁴ = 5⁷⁻⁴ = 5³
🔒 ❓ (iii) 9⁻⁷ ÷ 9⁴
📌 ✅ Answer:
⬥ 9⁻⁷ ÷ 9⁴ = 9⁻(7+4) = 9⁻¹¹
🔒 ❓ (iv) (13⁻²)⁻³
📌 ✅ Answer:
⬥ (13⁻²)⁻³ = 13⁶ = 13⁶
🔒 ❓ (v) m⁵n¹² (mn)⁹
📌 ✅ Answer:
⬥ (mn)⁹ = m⁹n⁹
⬥ m⁵n¹² × m⁹n⁹ = m¹⁴n²¹
🔒 ❓ Q6. If 12² = 144, what is
🔒 ❓ (i) (1.2)²
📌 ✅ Answer:
⬥ (1.2)² = (12/10)² = 144/100 = 1.44
🔒 ❓ (ii) (0.12)²
📌 ✅ Answer:
⬥ (0.12)² = (12/100)² = 144/10000 = 0.0144
🔒 ❓ (iii) (0.012)²
📌 ✅ Answer:
⬥ (0.012)² = (12/1000)² = 144/1000000 = 0.000144
🔒 ❓ (iv) 120²
📌 ✅ Answer:
⬥ 120² = (12×10)² = 12²×10² = 144×100 = 14400
🔒 ❓ Q7. Circle the numbers that are the same — 2⁴×3⁶, 6⁴×3², 6¹⁰, 18²×6², 6²⁴
📌 ✅ Answer:
🟢 Step 1
⬥ Express each in prime powers of 2 and 3
🔵 Step 2
⬥ 2⁴×3⁶ = 2⁴×3⁶
⬥ 6⁴×3² = (2⁴×3⁴)×3² = 2⁴×3⁶
⬥ 18²×6² = (2²×3⁴)×(2²×3²) = 2⁴×3⁶
⬥ 6¹⁰ = 2¹⁰×3¹⁰ (not same)
⬥ 6²⁴ = 2²⁴×3²⁴ (not same)
⬥ Same numbers are: 2⁴×3⁶, 6⁴×3², 18²×6²
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OTHER IMPORTANT QUESTIONS
🔹 PART A — MCQs
🔒 ❓ Question 1.
Which of the following expressions is equal to 6¹⁰?
🟢1️⃣ 2¹⁰ × 3¹⁰
🔵2️⃣ 2⁵ × 3⁵
🟡3️⃣ 6⁵ × 6⁵
🟣4️⃣ 18⁵
✔️ Answer: 🟢1️⃣
🔒 ❓ Question 2.
Which is greater without actual calculation?
🟢1️⃣ 5⁴
🔵2️⃣ 4⁵
🟡3️⃣ Both are equal
🟣4️⃣ Cannot be compared
✔️ Answer: 🔵2️⃣
🔒 ❓ Question 3.
Which expression represents exponential growth most accurately?
🟢1️⃣ n + 10
🔵2️⃣ 10n
🟡3️⃣ n²
🟣4️⃣ 2ⁿ
✔️ Answer: 🟣4️⃣
🔒 ❓ Question 4.
Which of the following is written in correct scientific notation?
🟢1️⃣ 30.81 × 10⁷
🔵2️⃣ 3.081 × 10⁸
🟡3️⃣ 0.3081 × 10⁹
🟣4️⃣ 3081 × 10⁵
✔️ Answer: 🔵2️⃣
🔒 ❓ Question 5.
Which pair of expressions are equal?
🟢1️⃣ 2⁴ × 3⁶ and 6⁴ × 3²
🔵2️⃣ 2⁶ × 3⁴ and 6⁸
🟡3️⃣ 18² × 6² and 6⁴
🟣4️⃣ 6¹² and 36⁵
✔️ Answer: 🟢1️⃣
🔒 ❓ Question 6.
Which statement is always true?
🟢1️⃣ Every cube is a square
🔵2️⃣ Every fourth power is a square
🟡3️⃣ Every square is a cube
🟣4️⃣ Every fifth power is a square
✔️ Answer: 🔵2️⃣
🔒 ❓ Question 7.
If a number is written as a¹², it is definitely a:
🟢1️⃣ Square only
🔵2️⃣ Cube only
🟡3️⃣ Both square and cube
🟣4️⃣ Neither square nor cube
✔️ Answer: 🟡3️⃣
🔒 ❓ Question 8.
Which is greater?
🟢1️⃣ 2¹⁰
🔵2️⃣ 10²
🟡3️⃣ Both equal
🟣4️⃣ Cannot be compared
✔️ Answer: 🟢1️⃣
🔒 ❓ Question 9.
How many digits are needed to form unique codes for about 9 billion items using digits 0–9?
🟢1️⃣ 9
🔵2️⃣ 10
🟡3️⃣ 11
🟣4️⃣ 8
✔️ Answer: 🔵2️⃣
🔒 ❓ Question 10.
Which expression simplifies to 10⁰?
🟢1️⃣ 10³ ÷ 10³
🔵2️⃣ 10² × 10²
🟡3️⃣ 10⁵ − 10⁵
🟣4️⃣ 10¹ ÷ 10²
✔️ Answer: 🟢1️⃣
🔹 PART B — Short Answer Questions
🔒 ❓ Question 11.
Explain why 2⁸ is greater than 8² without calculating their values.
📌 ✅ Answer:
🔹 8 = 2³, so 8² = 2⁶
🔹 Since 2⁸ > 2⁶, 2⁸ is greater
🔒 ❓ Question 12.
Why is exponential growth faster than additive growth?
📌 ✅ Answer:
🔹 Exponential growth multiplies repeatedly
🔹 Additive growth increases by a fixed amount only
🔒 ❓ Question 13.
Explain why 100² is much smaller than 2¹⁰⁰.
📌 ✅ Answer:
🔹 100² = 10⁴
🔹 2¹⁰⁰ grows exponentially and is far larger
🔒 ❓ Question 14.
Why is 0.5 × 10⁶ not in scientific notation?
📌 ✅ Answer:
🔹 The first factor must be ≥ 1
🔹 0.5 is less than 1
🔒 ❓ Question 15.
Explain why the product of two cube numbers is always a cube number.
📌 ✅ Answer:
🔹 Each cube has factors in groups of three
🔹 Their product still keeps factors in triples
🔒 ❓ Question 16.
Why is 10⁹ + 10⁹ written as 2 × 10⁹?
📌 ✅ Answer:
🔹 Both terms have the same power of 10
🔹 They can be added as coefficients
🔒 ❓ Question 17.
Explain why a⁻² is equal to 1/a².
📌 ✅ Answer:
🔹 Negative exponent means reciprocal
🔹 So a⁻² = 1/a²
🔒 ❓ Question 18.
Why is 64 both a square and a cube?
📌 ✅ Answer:
🔹 64 = 8² and also 4³
🔹 It satisfies both conditions
🔒 ❓ Question 19.
Explain why rewriting numbers in the same base helps comparison.
📌 ✅ Answer:
🔹 Same base allows direct exponent comparison
🔹 Larger exponent means larger value
🔒 ❓ Question 20.
Why are powers of 10 useful in estimating large quantities?
📌 ✅ Answer:
🔹 They show order of magnitude clearly
🔹 They simplify very large numbers
🔹 PART C — Detailed Answer Questions
🔒 ❓ Question 21.
Show that 18² × 6² is equal to 6⁴.
📌 ✅ Answer:
🔹 18 = 2 × 3²
🔹 18² × 6² = (2² × 3⁴) × (2² × 3²)
🔹 = 2⁴ × 3⁶
🔹 = (2 × 3)⁴ = 6⁴
🔒 ❓ Question 22.
Identify the greater number: 4³ or 3⁴. Explain logically.
📌 ✅ Answer:
🔹 4³ = (2²)³ = 2⁶
🔹 3⁴ = 81
🔹 Since 2⁶ = 64 < 81, 3⁴ is greater
🔒 ❓ Question 23.
Explain why every fourth power is always a square.
📌 ✅ Answer:
🔹 Fourth power = (n²)²
🔹 Square of a square is always a square
🔒 ❓ Question 24.
Show that q⁴⁶ (q prime) is both a fourth power and a sixth power.
📌 ✅ Answer:
🔹 46 = LCM of 2 and 3 × 2
🔹 q⁴⁶ = (q¹¹)⁴ and also (q⁷)⁶
🔒 ❓ Question 25.
Write 1928 as a product of powers in two different ways.
📌 ✅ Answer:
🔹 1928 = 2³ × 241
🔹 1928 = (2 × 241)¹ × 2²
🔒 ❓ Question 26.
Simplify (13⁻²)⁻³ and express the answer in exponential form.
📌 ✅ Answer:
🔹 (aᵐ)ⁿ = aᵐⁿ
🔹 (13⁻²)⁻³ = 13⁶
🔒 ❓ Question 27.
Explain why 10¹⁰ is ten times greater than 10⁹.
📌 ✅ Answer:
🔹 10¹⁰ ÷ 10⁹ = 10¹
🔹 Hence it is ten times greater
🔒 ❓ Question 28.
Estimate the total population if sheep ≈ 10⁹ and goats ≈ 10⁹.
📌 ✅ Answer:
🔹 Total = 10⁹ + 10⁹
🔹 = 2 × 10⁹
🔒 ❓ Question 29.
Calculate the total number of honeybees if there are 10⁸ colonies with 5 × 10⁴ bees each.
📌 ✅ Answer:
🔹 Total bees = 10⁸ × 5 × 10⁴
🔹 = 5 × 10¹²
🔒 ❓ Question 30.
Explain why scientific notation helps in comparing very large quantities.
📌 ✅ Answer:
🔹 It standardises numbers using powers of 10
🔹 Comparison becomes based on exponents
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