In this chapter, you will learn
- —Understand exponents and base in exponential notation
- —Apply laws of exponents to simplify expressions
- —Work with negative and zero exponents
- —Convert numbers to scientific notation
What are Exponents and Powers?
Exponent (or power) is a number that tells how many times the base is multiplied by itself.
Definition: In an, 'a' is the base and 'n' is the exponent. an = a × a × a × ... (n times)
Example: 23 = 2 × 2 × 2 = 8
Parts of Exponent:
- Base: The number being multiplied. In 34, the base is 3.
- Exponent: The number of times base is multiplied. In 34, the exponent is 4.
- Power: The result. 34 = 81 (the power is 81).
Examples:
- 52 = 25 (5 squared)
- 24 = 16 (2 to the power 4)
- 103 = 1000 (10 cubed)
Exam Tip
Know the difference between base and exponent. Always write the exponent as a superscript.
Common Mistake
Students write 2³ as 2×3 = 6. Correct: 2³ = 2×2×2 = 8, NOT 6.
Laws of Exponents
Laws of Exponents are rules for simplifying expressions with exponents.
Law 1: Product Rule - am × an = am+n
Example: 23 × 22 = 25 = 32
Law 2: Quotient Rule - am ÷ an = am-n
Example: 35 ÷ 32 = 33 = 27
Law 3: Power Rule - (am)n = am×n
Example: (22)3 = 26 = 64
Law 4: Product to Power - (a×b)n = an × bn
Example: (2×3)2 = 22 × 32 = 4 × 9 = 36
Exam Tip
These four laws are essential. Practice them until you can apply them instantly.
Common Mistake
a³ × a² ≠ a⁶. Correct: a³ × a² = a⁵ (add exponents when bases are same).
Zero and Negative Exponents
Zero Exponent: Any non-zero number to the power 0 equals 1.
Rule: a0 = 1 (where a ≠ 0)
Examples: 50 = 1, 1000 = 1, (-3)0 = 1
Negative Exponent: A negative exponent means reciprocal of the positive power.
Rule: a-n = 1/an
- 2-3 = 1/23 = 1/8
- 5-1 = 1/5
- 10-2 = 1/100 = 0.01
Exam Tip
Remember: a⁰ = 1 always. Negative exponent = take reciprocal.
Common Mistake
5⁻² ≠ -25. Correct: 5⁻² = 1/5² = 1/25 = 0.04
Scientific Notation
Scientific Notation expresses very large or very small numbers in a compact form using exponents.
Form: a × 10n, where 1 ≤ a < 10 and n is an integer
Examples:
- 5000 = 5 × 10³
- 0.00023 = 2.3 × 10⁻⁴
- 300,000,000 = 3 × 10⁸ (speed of light)
How to convert:
- Large numbers: Move decimal left, exponent is positive. 4500 = 4.5 × 10³
- Small numbers: Move decimal right, exponent is negative. 0.0032 = 3.2 × 10⁻³
Exam Tip
In scientific notation, the base (first number) must be between 1 and 10.
Common Mistake
0.0045 = 45 × 10⁻⁴ is wrong. Correct: 0.0045 = 4.5 × 10⁻³
Chapter Summary
Exponents and Powers simplify repeated multiplication. Key points:
- Exponent notation: an = a × a × ... (n times)
- Product rule: am × an = am+n
- Quotient rule: am ÷ an = am-n
- Power rule: (am)n = amn
- Special cases: a0 = 1, a-n = 1/an
- Scientific notation: Expresses large/small numbers as a × 10n
Exam Focus: Simplifying expressions using laws, negative/zero exponents, scientific notation conversions.