Power Play — Class 8 Maths | Preeti Kushwah Classes

📐 Introduction — What are Exponents?

Have you ever tried to write out the number 100,000,000 (one hundred million)? That is a lot of zeros! Mathematicians needed a shorter, more powerful way to write such large numbers. The solution is exponents — a compact notation that tells us how many times a number is multiplied by itself.

Instead of writing 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10, we simply write 10⁸. This reads as "10 raised to the power 8" or "10 to the 8th power." The exponent tells us the number of times the base appears as a factor.

📚 Key Terminology

⚡ Base

The number being multiplied repeatedly. In 5³, the base is 5.

🔼 Exponent (Power / Index)

The small number written above and to the right that tells how many times the base is multiplied by itself. In 5³, the exponent is 3.

✅ Power

The entire expression 5³ is called "5 raised to the power 3" or simply "5 cubed." The result, 125, is the value of the power.

🔢 Expanded Form

5³ = 5 × 5 × 5 = 125. The expanded form shows all the individual multiplications.

aⁿ = a × a × a × ... × a (n times) Here 'a' is the base and 'n' is the exponent (n is a positive integer)

📝 Reading Exponents Aloud

Expression	Read As	Expanded Form	Value
2⁴	"2 to the power 4" or "2 raised to 4"	2 × 2 × 2 × 2	16
3²	"3 squared"	3 × 3	9
7³	"7 cubed"	7 × 7 × 7	343
10⁵	"10 to the power 5"	10 × 10 × 10 × 10 × 10	100,000
(−4)²	"negative 4 squared"	(−4) × (−4)	16
(−2)³	"negative 2 cubed"	(−2) × (−2) × (−2)	−8

💡 Important: The exponent 2 is called "squared" (because a square of side a has area a²), and the exponent 3 is called "cubed" (because a cube of side a has volume a³).

💡 Why Do We Need Exponents?

🚀 Astronomy

The distance from the Earth to the Sun is approximately 150,000,000 km. Using exponents: 1.5 × 10⁸ km. Much neater!

🔬 Biology

A red blood cell is about 0.000007 m in diameter. In scientific notation: 7 × 10⁻⁶ m. Exponents help us express the very small too.

💻 Computing

Computer memory is measured in powers of 2. A kilobyte is 2¹⁰ = 1024 bytes. A gigabyte is 2³⁰ bytes.

🌎 Population

Earth has about 8,000,000,000 people. That is 8 × 10⁹. Exponents make big numbers manageable.

💡 Memory Aid: Think of the exponent as a "counter" — it counts how many times the base appears in the multiplication. Base = who, Exponent = how many times.

⚡ Exponents & Powers — Building Blocks

🔢 Negative Bases

When the base is negative, the sign of the result depends on whether the exponent is even or odd:

✅ Even Exponent → Positive Result

(−3)² = (−3) × (−3) = +9
(−2)⁴ = (−2) × (−2) × (−2) × (−2) = +16
The negatives cancel in pairs.

❌ Odd Exponent → Negative Result

(−3)³ = (−3) × (−3) × (−3) = −27
(−2)⁵ = −32
One negative is left unpaired.

💡 Careful! Note the difference: (−4)² = 16 but −4² = −16. In the first case, the negative is inside the bracket, so it gets squared. In the second, we square 4 first and then apply the negative sign.

📚 Powers of Small Numbers

Base	Power 1	Power 2	Power 3	Power 4	Power 5
2	2	4	8	16	32
3	3	9	27	81	243
5	5	25	125	625	3125
10	10	100	1,000	10,000	1,00,000

📋 Expressing Numbers as Powers of Prime Factors

Any positive integer can be expressed as a product of prime factors raised to powers. This is called the prime factorisation in exponential form.

Example: Express 360 as a product of prime powers.

Step 1: Divide by primes: 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2 × 2 × 2 × 3 × 15 = 2 × 2 × 2 × 3 × 3 × 5

Step 2: Write in exponential form: 360 = 2³ × 3² × 5¹

💡 Tip: Use a factor tree to break numbers down. Keep dividing by the smallest prime (2, 3, 5, 7, ...) until you reach 1. Then count how many times each prime appears and write it as an exponent.

📜 Laws of Exponents

The laws of exponents are powerful rules that let us simplify expressions involving powers. These laws work for any base (except zero where noted) and any integer exponents.

📑 Law 1: Product of Powers (Same Base)

a^m × aⁿ = a^m+n When multiplying powers with the same base, ADD the exponents

Why? Because a^m means m copies of a multiplied together, and aⁿ means n copies. Multiplying them gives us m + n copies in total.

Example 1: Simplify 2³ × 2⁴

2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128

Verification: 2³ = 8, 2⁴ = 16, and 8 × 16 = 128 = 2⁷ ✅

📑 Law 2: Quotient of Powers (Same Base)

a^m ÷ aⁿ = a^m−n (a ≠ 0) When dividing powers with the same base, SUBTRACT the exponents

Example 2: Simplify 5⁶ ÷ 5²

5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625

📑 Law 3: Power of a Power

(a^m)ⁿ = a^m×n When raising a power to another power, MULTIPLY the exponents

Example 3: Simplify (3²)⁴

(3²)⁴ = 3^2×4 = 3⁸ = 6561

Verification: 3² = 9, and 9⁴ = 9 × 9 × 9 × 9 = 6561 ✅

📑 Law 4: Power of a Product

(a × b)ⁿ = aⁿ × bⁿ The power distributes over multiplication

Example 4: Simplify (2 × 5)³

(2 × 5)³ = 2³ × 5³ = 8 × 125 = 1000

Or directly: (10)³ = 1000 ✅

📑 Law 5: Power of a Quotient

(a / b)ⁿ = aⁿ / bⁿ (b ≠ 0) The power distributes over division

📑 Law 6: Zero Exponent

a⁰ = 1 (a ≠ 0) Any non-zero number raised to the power zero equals 1

Why? Using the quotient law: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But any number divided by itself is 1. Therefore, a⁰ = 1.

✅ Examples of a⁰ = 1

5⁰ = 1
(−7)⁰ = 1
(1000)⁰ = 1
(3/4)⁰ = 1

❌ 0⁰ is Undefined

The expression 0⁰ is not defined in the NCERT context. The base must be non-zero for the zero exponent rule to apply.

📋 Summary of All Laws

Law	Rule	Example
Product of Powers	a^m × aⁿ = a^m+n	3² × 3⁵ = 3⁷
Quotient of Powers	a^m ÷ aⁿ = a^m−n	7⁸ ÷ 7³ = 7⁵
Power of a Power	(a^m)ⁿ = a^mn	(2³)² = 2⁶
Power of a Product	(ab)ⁿ = aⁿbⁿ	(3×4)² = 9×16
Power of a Quotient	(a/b)ⁿ = aⁿ/bⁿ	(2/5)³ = 8/125
Zero Exponent	a⁰ = 1 (a ≠ 0)	99⁰ = 1

💡 Memory Rhyme:
"Same base multiply? Add the powers high.
Same base divide? Subtract them — don't be shy.
Power on a power? Multiply the count.
Zero on top? The answer is one, no doubt!"

🔻 Negative Exponents

What happens when the exponent is a negative integer? The quotient law gives us the answer. Consider:

2³ ÷ 2⁵ = 2³⁻⁵ = 2⁻²

But we can also compute this directly: 2³ / 2⁵ = 8 / 32 = 1/4 = 1/2². So 2⁻² = 1/2².

a⁻ⁿ = 1 / aⁿ (a ≠ 0) A negative exponent means "take the reciprocal"

🔹 3⁻²

= 1/3² = 1/9

🔹 10⁻³

= 1/10³ = 1/1000 = 0.001

🔹 5⁻¹

= 1/5¹ = 1/5 = 0.2

🔹 (2/3)⁻²

= (3/2)² = 9/4

💡 Key Insight: A negative exponent does not make the number negative! It turns the number into a fraction (reciprocal). So 5⁻² = 1/25 (positive!), not −25.

📋 Reciprocal Rule for Fractions

When a fraction has a negative exponent, we simply flip the fraction and make the exponent positive:

(a/b)⁻ⁿ = (b/a)ⁿ Flip the fraction and change the sign of the exponent

Example 5: Simplify (3/7)⁻²

(3/7)⁻² = (7/3)² = 49/9 = 5 4/9

🔢 Combining Laws with Negative Exponents

Example 6: Simplify 2³ × 2⁻⁵ × 2⁴

Step 1: Same base (2), so add exponents: 2^{3 + (−5) + 4}

Step 2: 3 + (−5) + 4 = 3 − 5 + 4 = 2

Answer: 2² = 4

💡 Quick Rule: Negative exponent = "move across the fraction bar." A factor with a negative exponent in the numerator moves to the denominator (with a positive exponent), and vice versa. Example: x⁻³/y⁻² = y²/x³.

🔢 Powers of 10

Powers of 10 are especially useful because our number system is based on 10 (the decimal system). Each power of 10 corresponds to a specific place value.

📊 Positive Powers of 10

Power	Value	Name	Number of Zeros
10⁰	1	One	0
10¹	10	Ten	1
10²	100	Hundred	2
10³	1,000	Thousand	3
10⁴	10,000	Ten Thousand	4
10⁵	1,00,000	Lakh	5
10⁶	10,00,000	Ten Lakh / Million	6
10⁷	1,00,00,000	Crore	7
10⁸	10,00,00,000	Ten Crore	8
10⁹	1,00,00,00,000	Arab / Billion	9

💡 Pattern: 10ⁿ is 1 followed by n zeros. So 10⁶ is 1 followed by 6 zeros = 1,000,000.

📊 Negative Powers of 10

Power	Value (Fraction)	Value (Decimal)
10⁻¹	1/10	0.1
10⁻²	1/100	0.01
10⁻³	1/1,000	0.001
10⁻⁴	1/10,000	0.0001
10⁻⁵	1/1,00,000	0.00001
10⁻⁶	1/10,00,000	0.000001

💡 Pattern for Negative Powers: 10⁻ⁿ has the digit 1 in the n-th decimal place. So 10⁻⁴ = 0.0001 (the 1 is in the 4th place after the decimal).

🔬 Standard Form (Scientific Notation)

Scientists, engineers, and mathematicians use standard form (also called scientific notation) to express very large or very small numbers compactly. A number is in standard form when it is written as:

a × 10ⁿ where 1 ≤ a < 10 and n is an integer

The number a is called the coefficient (or significand) and must be at least 1 but less than 10. The power of 10 tells us the order of magnitude.

🔼 Converting Large Numbers to Standard Form

To convert a large number to standard form, move the decimal point to the left until you have a number between 1 and 10, then count how many places you moved.

Example 7: Express 384,400,000 (distance to the Moon in metres) in standard form.

Step 1: Place the decimal after the first non-zero digit: 3.844

Step 2: Count how many places we moved the decimal: 8 places to the left.

Step 3: Write in standard form: 3.844 × 10⁸ m

🔽 Converting Small Numbers to Standard Form

For very small numbers, move the decimal point to the right until you have a number between 1 and 10. The number of places moved becomes a negative exponent.

Example 8: Express 0.000000016 (charge of an electron in coulombs, approximately) in standard form.

Step 1: Move the decimal right to get 1.6 (a number between 1 and 10).

Step 2: We moved the decimal 8 places to the right.

Step 3: Write in standard form: 1.6 × 10⁻⁸ (approximation)

📊 Real-World Numbers in Standard Form

Quantity	Value	Standard Form
Distance to the Sun	150,000,000 km	1.5 × 10⁸ km
Speed of light	300,000,000 m/s	3 × 10⁸ m/s
Diameter of Earth	12,742,000 m	1.2742 × 10⁷ m
Mass of Earth	5,970,000,000,000,000,000,000,000 kg	5.97 × 10²⁴ kg
Size of a hydrogen atom	0.00000000012 m	1.2 × 10⁻¹⁰ m
Mass of an electron	0.000000000000000000000000000000911 kg	9.11 × 10⁻³¹ kg

📋 Comparing Numbers Using Standard Form

Standard form makes it easy to compare very large or very small numbers. Simply compare the powers of 10 first. If the powers are the same, compare the coefficients.

⚡ Which is larger?

3.2 × 10¹² vs 8.5 × 10¹⁰

Answer: 10¹² > 10¹⁰, so 3.2 × 10¹² is larger (by about 100 times).

⚡ Which is larger?

4.7 × 10⁵ vs 9.1 × 10⁵

Answer: Same power of 10. Compare coefficients: 9.1 > 4.7, so 9.1 × 10⁵ is larger.

💡 Quick Check: Is your standard form correct?
✅ The coefficient must be ≥ 1 and < 10 (i.e., one non-zero digit before the decimal point).
✅ Large numbers have a positive exponent.
✅ Small numbers (between 0 and 1) have a negative exponent.

✏️ NCERT-Style Worked Examples

Problem 1: Simplify and express the result in exponential form: 2⁵ × 2³ × 2⁻⁴

Step 1: Same base (2), so add the exponents.

Step 2: 2^5+3+(−4) = 2⁴

Answer: 2⁴ = 16

Problem 2: Find the value of (3²)³ ÷ 3⁴

Step 1: Apply Power of a Power: (3²)³ = 3^2×3 = 3⁶

Step 2: Apply Quotient law: 3⁶ ÷ 3⁴ = 3⁶⁻⁴ = 3²

Answer: 3² = 9

Problem 3: Express 0.00000459 in standard form.

Step 1: Move the decimal right to get 4.59 (between 1 and 10).

Step 2: We moved 6 places to the right, so exponent is −6.

Answer: 4.59 × 10⁻⁶

Problem 4: Simplify (5/8)⁻² × (5/8)⁵

Step 1: Same base (5/8), add exponents: (5/8)⁻²⁺⁵ = (5/8)³

Step 2: Calculate: (5/8)³ = 5³/8³ = 125/512

Answer: 125/512

Problem 5: Find the value of x if 3^x = 729.

Step 1: Express 729 as a power of 3.

Step 2: 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81, 3⁵ = 243, 3⁶ = 729

Step 3: So 3^x = 3⁶, which gives x = 6.

Problem 6: The mass of a dust particle is about 7.53 × 10⁻¹⁰ kg. The mass of the Earth is about 5.97 × 10²⁴ kg. How many times heavier is the Earth than the dust particle?

Step 1: Divide: (5.97 × 10²⁴) ÷ (7.53 × 10⁻¹⁰)

Step 2: = (5.97 / 7.53) × 10^24−(−10) = 0.793 × 10³⁴

Step 3: Convert to proper standard form: 7.93 × 10³³

Answer: The Earth is approximately 7.93 × 10³³ times heavier than the dust particle.

Problem 7: Simplify: (2³ × 3² × 5) / (2⁵ × 3 × 5²)

Step 1: Group same bases: (2³/2⁵) × (3²/3¹) × (5¹/5²)

Step 2: Apply quotient law for each base:
= 2³⁻⁵ × 3²⁻¹ × 5¹⁻²
= 2⁻² × 3¹ × 5⁻¹

Step 3: = (1/4) × 3 × (1/5) = 3/20

Answer: 3/20 or 0.15

Problem 8: Express 72,000,000 as a product of a number between 1 and 10 multiplied by a power of 10. Also express 0.000038 in the same form.

Part (a): 72,000,000 = 7.2 × 10,000,000 = 7.2 × 10⁷

Part (b): 0.000038 = 3.8 × 0.00001 = 3.8 × 10⁻⁵ = 3.8 × 10⁻⁵

📊 Chapter Summary

📋 All Formulas at a Glance

a^m × aⁿ = a^m+n Product of Powers

a^m ÷ aⁿ = a^m−n Quotient of Powers (a ≠ 0)

(a^m)ⁿ = a^mn Power of a Power

a⁰ = 1 & a⁻ⁿ = 1/aⁿ Zero Exponent & Negative Exponent (a ≠ 0)

Standard Form: a × 10ⁿ (1 ≤ a < 10) Used for very large and very small numbers

⚠️ Common Mistakes to Avoid

❌ Adding exponents of different bases

2³ × 3² ≠ 6⁵. The product law only works when the bases are the same.

❌ Confusing (−a)ⁿ with −aⁿ

(−3)² = 9 but −3² = −9. The brackets make all the difference!

❌ Thinking negative exponent = negative number

2⁻³ = 1/8 = 0.125 (positive!). A negative exponent gives a reciprocal, not a negative number.

❌ Coefficient out of range in standard form

45.3 × 10⁶ is not in standard form because 45.3 is not between 1 and 10. Correct form: 4.53 × 10⁷.

📚 Key Terms Recap

Term	Definition
Base	The number being raised to a power (e.g., in 5³, base = 5)
Exponent	The power to which the base is raised (e.g., in 5³, exponent = 3)
Power	The expression aⁿ or its value
Standard Form	a × 10ⁿ where 1 ≤ a < 10
Product Law	a^m × aⁿ = a^m+n
Quotient Law	a^m ÷ aⁿ = a^m−n
Zero Exponent	a⁰ = 1 for any a ≠ 0
Negative Exponent	a⁻ⁿ = 1/aⁿ (reciprocal)

💡 Exam Day Checklist:
✅ Read the question — is it asking you to simplify, evaluate, or convert to standard form?
✅ Write the relevant law of exponents before applying it.
✅ Check: are the bases the same before adding/subtracting exponents?
✅ For standard form, ensure the coefficient is between 1 and 10.
✅ Show all steps clearly — marks are given for working.

🧠 Multiple Choice Questions (10 MCQs)

Select one option for each question, then click Check Score to see how you did!

Q1. The value of 2⁵ is:
- a) 10
- b) 32
- c) 25
- d) 64
✅ Answer: (b) 32 — 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
Q2. 3⁴ × 3² equals:
- a) 3⁸
- b) 3⁶
- c) 9⁶
- d) 3²
✅ Answer: (b) 3⁶ — Same base, add exponents: 3⁴⁺² = 3⁶ = 729.
Q3. The value of 7⁰ is:
- a) 0
- b) 7
- c) 1
- d) Undefined
✅ Answer: (c) 1 — Any non-zero number raised to the power 0 equals 1.
Q4. 5⁻² is equal to:
- a) −25
- b) −10
- c) 1/25
- d) 25
✅ Answer: (c) 1/25 — 5⁻² = 1/5² = 1/25.
Q5. Express 45,000,000 in standard form:
- a) 45 × 10⁶
- b) 4.5 × 10⁷
- c) 4.5 × 10⁶
- d) 0.45 × 10⁸
✅ Answer: (b) 4.5 × 10⁷ — 45,000,000 = 4.5 × 10⁷. The coefficient must be between 1 and 10.
Q6. (2³)² equals:
- a) 2⁵
- b) 2⁶
- c) 2⁹
- d) 4⁶
✅ Answer: (b) 2⁶ — Power of a power: (2³)² = 2^3×2 = 2⁶ = 64.
Q7. (−1)¹⁰⁰ equals:
- a) 1
- b) −1
- c) 100
- d) 0
✅ Answer: (a) 1 — Since 100 is even, (−1)¹⁰⁰ = 1.
Q8. Which is the standard form of 0.0000062?
- a) 62 × 10⁻⁷
- b) 6.2 × 10⁻⁶
- c) 6.2 × 10⁶
- d) 0.62 × 10⁻⁵
✅ Answer: (b) 6.2 × 10⁻⁶ — Move decimal 6 places right to get 6.2, hence ×10⁻⁶.
Q9. 10⁸ ÷ 10⁵ equals:
- a) 10⁴⁰
- b) 10³
- c) 10¹³
- d) 1³
✅ Answer: (b) 10³ — Same base, subtract exponents: 10⁸⁻⁵ = 10³ = 1000.
Q10. If 2^x = 64, then x equals:
- a) 4
- b) 5
- c) 6
- d) 8
✅ Answer: (c) 6 — 2⁶ = 64, so x = 6.

📚 NCERT Questions & Answers

✏️ Fill in the Blanks

1. a^m × aⁿ = a^____ ▼

a^m+n — When multiplying powers with the same base, add the exponents.

2. (a^m)ⁿ = a^____ ▼

a^mn — Power of a power: multiply the exponents.

3. 10⁻³ = ____ ▼

1/1000 or 0.001 — 10⁻³ = 1/10³ = 1/1000.

4. In standard form, 56,000 = ____ ▼

5.6 × 10⁴ — Move decimal 4 places left to get 5.6.

5. Any non-zero number raised to the power zero equals ____ ▼

1 — a⁰ = 1 for all a ≠ 0.

✅ True or False

1. 2³ × 3² = 6⁵ ▼

False. The product law (add exponents) only applies when the bases are the same. Here the bases are 2 and 3 (different). 2³ × 3² = 8 × 9 = 72, while 6⁵ = 7776.

2. (−5)⁰ = 1 ▼

True. Any non-zero number (including negative numbers) raised to the power 0 equals 1.

3. 10⁻² is a negative number. ▼

False. 10⁻² = 1/100 = 0.01, which is a positive number. A negative exponent gives a reciprocal, not a negative number.

4. (−2)⁵ = −32 ▼

True. Since the exponent is odd, the result is negative. (−2)⁵ = (−2) × (−2) × (−2) × (−2) × (−2) = −32.

5. 3.45 × 10³ = 3450 ▼

True. 3.45 × 10³ = 3.45 × 1000 = 3450.

✍️ Short Answer Questions

Q1. Explain why a⁰ = 1 using the quotient law of exponents.

By the quotient law, aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But any non-zero number divided by itself equals 1. So a⁰ = 1.
Q2. Simplify: (4³ × 4⁻⁵) / 4⁻⁴

Numerator: 4³⁺⁽⁻⁵⁾ = 4⁻². Now divide: 4⁻² / 4⁻⁴ = 4^{−2−(−4)} = 4² = 16.
Q3. Express 0.000567 in standard form.

Move the decimal 4 places to the right to get 5.67. So 0.000567 = 5.67 × 10⁻⁴.
Q4. Write 3.2 × 10⁵ in the usual form (expanded number).

3.2 × 10⁵ = 3.2 × 100,000 = 320,000.
Q5. Find the value of (−1)⁵⁰ + (−1)⁵¹.

(−1)⁵⁰ = 1 (even power), (−1)⁵¹ = −1 (odd power). Sum = 1 + (−1) = 0.

📖 Long Answer Questions

Q1. State and prove the product law of exponents. Use it to simplify: 5³ × 5⁴ × 5⁻².

Product Law: a^m × aⁿ = a^m+n

Proof: a^m means a multiplied by itself m times. Similarly, aⁿ means a multiplied n times. When we multiply these together, we get a multiplied (m + n) times in total.
a^m × aⁿ = (a × a × ... m times) × (a × a × ... n times) = a × a × ... (m+n) times = a^m+n. ✅

Application:
5³ × 5⁴ × 5⁻² = 5^3+4+(−2) = 5⁵ = 3125.

Q2. A bacterium divides into 2 every hour. Starting with 1 bacterium, how many bacteria will there be after 10 hours? Express the answer as a power and in standard form. If each bacterium weighs about 9.5 × 10⁻¹³ g, find the total mass after 10 hours.

Number of bacteria:
After 1 hour: 2¹ = 2
After 2 hours: 2² = 4
After n hours: 2ⁿ
After 10 hours: 2¹⁰ = 1024 bacteria.

Standard form: 1024 = 1.024 × 10³.

Total mass:
= 1024 × 9.5 × 10⁻¹³ g
= 1.024 × 10³ × 9.5 × 10⁻¹³ g
= (1.024 × 9.5) × 10^3+(−13) g
= 9.728 × 10⁻¹⁰ g
The total mass is approximately 9.73 × 10⁻¹⁰ g (less than a billionth of a gram!).

Q3. Explain negative exponents with examples. Show that (a/b)⁻ⁿ = (b/a)ⁿ.

Negative Exponent Rule: a⁻ⁿ = 1/aⁿ (for a ≠ 0).

Examples:
• 4⁻¹ = 1/4 = 0.25
• 2⁻³ = 1/2³ = 1/8 = 0.125
• 10⁻⁴ = 1/10,000 = 0.0001

Proof of (a/b)⁻ⁿ = (b/a)ⁿ:
(a/b)⁻ⁿ = 1 / (a/b)ⁿ   [by negative exponent rule]
= 1 / (aⁿ/bⁿ)   [by power of a quotient]
= bⁿ / aⁿ   [taking reciprocal]
= (b/a)ⁿ   [by power of a quotient in reverse]

Example: (2/5)⁻³ = (5/2)³ = 125/8 = 15.625 ✅

Q4. The distance between the Sun and Neptune is about 4.5 × 10⁹ km. Light travels at about 3 × 10⁵ km/s. How long does light take to travel from the Sun to Neptune? Express the answer in standard form and also in minutes.

Time = Distance / Speed:
= (4.5 × 10⁹) / (3 × 10⁵)
= (4.5 / 3) × 10⁹⁻⁵
= 1.5 × 10⁴ seconds
= 15,000 seconds

In minutes: 15,000 / 60 = 250 minutes (or about 4 hours 10 minutes).

So sunlight takes approximately 250 minutes to reach Neptune!

Q5. Simplify the following using laws of exponents:
(i) (2⁵ × 3² × 7) / (2³ × 3⁴)
(ii) [(5⁻¹ × 3²) / (5³ × 3⁻²)]²

(i) Group by base:
= (2⁵/2³) × (3²/3⁴) × 7
= 2⁵⁻³ × 3²⁻⁴ × 7
= 2² × 3⁻² × 7
= 4 × (1/9) × 7 = 28/9 = 3 1/9

(ii) Simplify inside the brackets first:
= [(5⁻¹/5³) × (3²/3⁻²)]²
= [5⁻¹⁻³ × 3²⁻⁽⁻²⁾]²
= [5⁻⁴ × 3⁴]²
= 5⁻⁸ × 3⁸
= 3⁸ / 5⁸
= (3/5)⁸
= 6561 / 390625 = (3/5)⁸

🌟 Fun Facts & Did You Know?

📚 Googol — A Famous Exponent

A googol is 10¹⁰⁰ — the number 1 followed by 100 zeros. It was named by a 9-year-old boy, Milton Sirotta, nephew of mathematician Edward Kasner. Google (the company) got its name from a misspelling of "googol"!

🇮🇳 Ancient Indian Mathematics

Indian mathematicians used large numbers with ease. The Lalitavistara Sutra (a Buddhist text) describes a number system going up to 10⁵³! Long before Western mathematics had names for such quantities, Indian scholars were thinking on a cosmic scale.

💻 Binary — Powers of 2

Computers use the binary system (base 2). Everything inside a computer is stored as powers of 2. A byte = 2³ bits = 8 bits. A kilobyte = 2¹⁰ = 1024 bytes. A terabyte = 2⁴⁰ bytes!

🔬 Powers in Science

The pH scale in chemistry uses powers of 10. A pH of 3 means an H⁺ concentration of 10⁻³ mol/L. Each unit change in pH represents a 10-fold change in acidity!

🌟 Stars in the Universe

There are estimated to be about 10²⁴ stars in the observable universe. That is 1,000,000,000,000,000,000,000,000 stars! Without exponents, even writing this number would fill up your notebook.

💉 Atoms are Tiny

The diameter of an atom is about 10⁻¹⁰ metres (0.0000000001 m). If you lined up 10 billion atoms side by side, they would only span about 1 metre. Exponents are the only practical way to describe such small sizes.

🌊 Rice on a Chessboard

There is an ancient legend where a sage asks a king for rice: 1 grain on the first square of a chessboard, 2 on the second, 4 on the third, and so on — doubling each time. The total is 2⁶⁴ − 1 = about 1.8 × 10¹⁹ grains. That is more rice than the entire world produces in centuries!

🚀 Rocket Speed

The escape velocity from Earth is about 1.12 × 10⁴ m/s (about 40,320 km/h). ISRO's PSLV rockets must reach this speed to send satellites into orbit. Powers of 10 help scientists calculate these enormous speeds precisely.

💡 Think About It: From the tiniest atom to the farthest galaxy, exponents are the language of scale. They let us grasp the ungraspable — numbers so large or so small that writing them out in full would be impractical. Mastering exponents is like gaining a superpower for mathematics and science!

🃏 Quick Revision Flashcards

Use these flashcards for quick revision before your exam. Read the question, answer mentally, then check!

Q: What is the base in 7⁴?

A: The base is 7.

Q: 10⁰ = ?

A: 1 (any non-zero number to the power 0 is 1).

Q: 2⁻⁴ = ?

A: 1/2⁴ = 1/16.

Q: Product law?

A: a^m × aⁿ = a^m+n

Q: Quotient law?

A: a^m ÷ aⁿ = a^m−n

Q: Power of a power?

A: (a^m)ⁿ = a^mn

Q: 3 × 10⁴ = ?

A: 30,000

Q: Standard form of 0.00045?

A: 4.5 × 10⁻⁴

Q: (−1)^odd = ?

A: −1

Q: (2/3)⁻¹ = ?

A: 3/2 (flip the fraction)

Q: How many zeros in 10⁶?

A: 6 zeros → 1,000,000

Q: Is 15 × 10³ in standard form?

A: No! 15 is not between 1 and 10. Correct: 1.5 × 10⁴.

💡 Final Revision Mantra:
• Same base, multiply → Add exponents.
• Same base, divide → Subtract exponents.
• Power of a power → Multiply exponents.
• Zero exponent → Always 1 (a ≠ 0).
• Negative exponent → Take the reciprocal.
• Standard form → a × 10ⁿ where 1 ≤ a < 10.

You've got this! Go ace that exam! 💪

⚡ Power Play

⚡ Base

🔼 Exponent (Power / Index)

✅ Power

🔢 Expanded Form

🚀 Astronomy

🔬 Biology

💻 Computing

🌎 Population

✅ Even Exponent → Positive Result

❌ Odd Exponent → Negative Result

Example: Express 360 as a product of prime powers.

Example 1: Simplify 23 × 24

Example 2: Simplify 56 ÷ 52

Example 3: Simplify (32)4

Example 4: Simplify (2 × 5)3

✅ Examples of a0 = 1

❌ 00 is Undefined

🔹 3−2

🔹 10−3

🔹 5−1

🔹 (2/3)−2

Example 5: Simplify (3/7)−2

Example 6: Simplify 23 × 2−5 × 24

Example 7: Express 384,400,000 (distance to the Moon in metres) in standard form.

Example 8: Express 0.000000016 (charge of an electron in coulombs, approximately) in standard form.

⚡ Which is larger?

⚡ Which is larger?

Problem 1: Simplify and express the result in exponential form: 25 × 23 × 2−4

Problem 2: Find the value of (32)3 ÷ 34

Problem 3: Express 0.00000459 in standard form.

Problem 4: Simplify (5/8)−2 × (5/8)5

Problem 5: Find the value of x if 3x = 729.

Problem 6: The mass of a dust particle is about 7.53 × 10−10 kg. The mass of the Earth is about 5.97 × 1024 kg. How many times heavier is the Earth than the dust particle?

Problem 7: Simplify: (23 × 32 × 5) / (25 × 3 × 52)

Problem 8: Express 72,000,000 as a product of a number between 1 and 10 multiplied by a power of 10. Also express 0.000038 in the same form.

❌ Adding exponents of different bases

❌ Confusing (−a)n with −an

❌ Thinking negative exponent = negative number

❌ Coefficient out of range in standard form

📚 Googol — A Famous Exponent

🇮🇳 Ancient Indian Mathematics

💻 Binary — Powers of 2

🔬 Powers in Science

🌟 Stars in the Universe

💉 Atoms are Tiny

🌊 Rice on a Chessboard

🚀 Rocket Speed

Q: What is the base in 74?

Q: 100 = ?

Q: 2−4 = ?

Q: Product law?

Q: Quotient law?

Q: Power of a power?

Q: 3 × 104 = ?

Q: Standard form of 0.00045?

Q: (−1)odd = ?

Q: (2/3)−1 = ?

Q: How many zeros in 106?

Q: Is 15 × 103 in standard form?

📤 Share This Page

Example 1: Simplify 2³ × 2⁴

Example 2: Simplify 5⁶ ÷ 5²

Example 3: Simplify (3²)⁴

Example 4: Simplify (2 × 5)³

✅ Examples of a⁰ = 1

❌ 0⁰ is Undefined

🔹 3⁻²

🔹 10⁻³

🔹 5⁻¹

🔹 (2/3)⁻²

Example 5: Simplify (3/7)⁻²

Example 6: Simplify 2³ × 2⁻⁵ × 2⁴

Problem 1: Simplify and express the result in exponential form: 2⁵ × 2³ × 2⁻⁴

Problem 2: Find the value of (3²)³ ÷ 3⁴

Problem 4: Simplify (5/8)⁻² × (5/8)⁵

Problem 5: Find the value of x if 3^x = 729.

Problem 6: The mass of a dust particle is about 7.53 × 10⁻¹⁰ kg. The mass of the Earth is about 5.97 × 10²⁴ kg. How many times heavier is the Earth than the dust particle?

Problem 7: Simplify: (2³ × 3² × 5) / (2⁵ × 3 × 5²)

❌ Confusing (−a)ⁿ with −aⁿ

Q: What is the base in 7⁴?

Q: 10⁰ = ?

Q: 2⁻⁴ = ?

Q: 3 × 10⁴ = ?

Q: (−1)^odd = ?

Q: (2/3)⁻¹ = ?

Q: How many zeros in 10⁶?

Q: Is 15 × 10³ in standard form?