Learning Objectives
- Understand negative exponents and their meaning
- Apply laws of exponents to expressions with integer exponents
- Express numbers in standard form (scientific notation)
- Compare very large and very small numbers using powers of 10
Key Concepts
Negative Exponents
For any non-zero number a: a⁻ⁿ = 1/aⁿ. A negative exponent means the reciprocal of the positive exponent.
Examples: 2⁻³ = 1/2³ = 1/8. 10⁻⁴ = 1/10000 = 0.0001.
Laws of Exponents
These laws apply for any non-zero base a and integers m, n:
- Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
- Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power of a Power: (aᵐ)ⁿ = aᵐˣⁿ
- Product of Powers: aᵐ × bᵐ = (ab)ᵐ
- Quotient of Powers: aᵐ ÷ bᵐ = (a/b)ᵐ
- Zero Exponent: a⁰ = 1 (for a ≠ 0)
Standard Form (Scientific Notation)
A number is in standard form when it is written as k × 10ⁿ, where 1 ≤ k < 10 and n is an integer.
Large numbers: 300000000 = 3 × 10⁸. Small numbers: 0.00025 = 2.5 × 10⁻⁴.
Comparing Numbers in Standard Form
To compare numbers in standard form, first compare the powers of 10. If powers are equal, compare the values of k.
Summary
Exponents with negative integers represent reciprocals. Laws of exponents extend to all integer exponents. Standard form (scientific notation) is used to express very large or very small numbers conveniently. Any non-zero number raised to the power 0 equals 1.
Important Terms
- Exponent: The power to which a number is raised
- Base: The number being raised to a power
- Negative Exponent: Indicates the reciprocal: a⁻ⁿ = 1/aⁿ
- Standard Form: A number expressed as k × 10ⁿ where 1 ≤ k < 10
- Scientific Notation: Another name for standard form
Quick Revision
- a⁻ⁿ = 1/aⁿ; a⁰ = 1 (a ≠ 0)
- aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- Standard form: k × 10ⁿ (1 ≤ k < 10)
- Very large numbers have positive n; very small numbers have negative n
- Use laws of exponents to simplify expressions before calculating