Chapter 4 - Exponents and Powers Revision Notes - Class 7 Mathematics

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📌 Key Points

In aⁿ, 'a' is the base and 'n' is the exponent. aⁿ means a multiplied by itself n times.
2³ = 2 × 2 × 2 = 8 (not 2 × 3 = 6). The exponent tells how many times to multiply.
Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents when bases are same).
Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract exponents when dividing same base).
Power rule: (aᵐ)ⁿ = aᵐˣⁿ (multiply exponents when power is raised to another power).
Zero exponent: a⁰ = 1 for any non-zero number a.
Negative exponent: a⁻ⁿ = 1/aⁿ (take reciprocal of the positive power).
Product to power: (a × b)ⁿ = aⁿ × bⁿ (distribute exponent to each factor).
Same base, different exponents: 2² = 4, 2³ = 8, 2⁴ = 16 (powers increase exponentially).
Scientific notation: a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
Large numbers: 5000 = 5 × 10³ (move decimal left = positive exponent).
Small numbers: 0.0032 = 3.2 × 10⁻³ (move decimal right = negative exponent).
Parentheses matter: (-2)⁴ = 16 but -2⁴ = -16 (exponent applies only to 2).
Odd vs even powers: (-3)³ = -27 (odd), (-3)⁴ = 81 (even).
1 to any power = 1. 0 to any positive power = 0.

📘 Important Definitions

Exponent (Power)

A number that indicates how many times the base is multiplied by itself.

Base

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

Exponential Form

Writing a number as aⁿ. Example: 8 = 2³ is exponential form.

Expanded Form

Writing multiplication out: 2³ = 2 × 2 × 2 (expanded).

Scientific Notation

Form a × 10ⁿ for very large or very small numbers, where 1 ≤ a < 10.

Negative Exponent

a⁻ⁿ represents the reciprocal of aⁿ, which is 1/aⁿ.

Zero Exponent

For any non-zero number a, a⁰ = 1.

Laws of Exponents

Rules for simplifying expressions with exponents (product, quotient, power rules).

🔢 Formulas & Laws

Product Rule

aᵐ × aⁿ = aᵐ⁺ⁿ

Example: 3² × 3³ = 3⁵

Quotient Rule

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Example: 5⁴ ÷ 5² = 5²

Power Rule

(aᵐ)ⁿ = aᵐˣⁿ

Example: (2²)³ = 2⁶

Product to Power

(a × b)ⁿ = aⁿ × bⁿ

Example: (2 × 3)² = 4 × 9 = 36

Negative Exponent

a⁻ⁿ = 1/aⁿ

Example: 2⁻³ = 1/8

⚠️ Common Mistakes

✗ Wrong: 2³ = 2 × 3 = 6

✓ Correct: Correct: 2³ = 2 × 2 × 2 = 8. Exponent means multiply, not add.

✗ Wrong: 3² × 3² = 3⁴ is wrong because 9 × 9 = 81, not 81

✓ Correct: Actually correct! 3² × 3² = 3⁴ = 81. Using product rule is right.

✗ Wrong: 5⁻² = -25

✓ Correct: Correct: 5⁻² = 1/25 = 0.04. Negative exponent = reciprocal, not negative sign.

✗ Wrong: (-2)² = -4

✓ Correct: Correct: (-2)² = 4. Even power of negative number is positive.

✗ Wrong: 0.0045 = 45 × 10⁻⁴

✓ Correct: Correct: 0.0045 = 4.5 × 10⁻³. In scientific notation, first number must be 1-10.

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Simplify expressions using laws of exponents

2m★★★

Product and quotient rules with same base

1m★★★

Power rule and nested exponents

2m★★

Zero and negative exponents

1m★★

Scientific notation conversions

2m★★

Evaluating expressions with multiple bases and exponents

🎯 Last-Minute Recall