Algebra · StudentHub Lesson
Exponents and the Laws of Indices
Apply the laws of exponents to simplify expressions with powers.
20 minIntermediate
01
What you will learn
- Understand exponents as repeated multiplication
- Apply the product rule for exponents
- Apply the quotient and power rules
- Simplify expressions with negative and zero exponents
- Evaluate numerical expressions with exponents
02
Watch the lesson
Algebra Basics: Laws Of Exponents - Math Antics · mathantics
Watch on YouTube03
Topic notes
Main Idea
Exponents (indices) show repeated multiplication of a base number, and specific laws let us simplify expressions involving powers efficiently.
Key Concepts
- Product rule: multiplying the same base adds exponents
- Quotient rule: dividing the same base subtracts exponents
- Power rule: raising a power to a power multiplies exponents
- Any nonzero number to the power 0 equals 1
- Negative exponents represent reciprocals
Definitions
- Base: the number being multiplied
- Exponent/Index: how many times the base is multiplied by itself
- Reciprocal: 1 divided by a number
Formulas
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (a≠0)
- a⁻ⁿ = 1/aⁿ
Examples
- 2³ × 2⁴ = 2⁷ = 128
- x⁵ ÷ x² = x³
- 3⁻² = 1/9
Common Mistakes
- Adding exponents when the bases are different
- Forgetting the a⁰=1 rule
- Mishandling negative exponents (forgetting the reciprocal)
04
Key concepts
Repeated multiplicationProduct ruleQuotient rulePower ruleZero and negative exponents
05
Important terms
- Base
- The number being raised to a power.
- Exponent
- The number indicating repeated multiplication of the base.
- Reciprocal
- 1 divided by a given number.
06
Worked examples
Problem
Simplify 5² × 5³
- 1. Add exponents since bases match
- 2. 5²⁺³=5⁵
Answer: 3125
Problem
Simplify x⁷ ÷ x³
- 1. Subtract exponents since bases match
- 2. x⁷⁻³
Answer: x⁴
07
Quick revision
- Same base multiply: add exponents
- Same base divide: subtract exponents
- Power of a power: multiply exponents
- Anything to the power 0 is 1
- Negative exponent means reciprocal
- Rules only apply to the same base
08
Check your understanding
Question 1 · Multiple choice
Simplify 3⁴ × 3²
Question 2 · Multiple choice
What is 7⁰?
Question 3 · Short answer
Simplify 2⁻³.
Question 4 · Short answer
Simplify (x³)⁴.
Question 5 · True or false
aᵐ × bⁿ can be simplified by adding exponents when a and b are different bases.
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