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Unit: Basic Algebra
Chapter: Expressions Of Exponents & Powers
Reference: – What is an Exponent, Base and Exponent (Power), Reading and Writing Exponents, Exponential Form vs Expanded Form, Square and Cube, Special Exponents (0, 1), First Law of Exponents (Product Rule), Second Law of Exponents (Quotient Rule), Third Law of Exponents (Power of a Power), Fourth Law of Exponents (Power of a Product), Fifth Law of Exponents (Power of a Quotient), Zero Exponent Rule, Negative Exponents, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is an Exponent and What It Represents
- How to Read and Write Numbers in Exponential Form
- Basic Laws of Exponents
- Special Cases: Exponent 0 and Exponent 1
- Meaning of Negative Exponents
Introduction to Exponents & Powers
Definition
An exponent tells us how many times a number (called the base) is multiplied by itself. It is written as a small number to the upper right of the base. For example, in 5³, 5 is the base and 3 is the exponent, meaning 5 × 5 × 5 = 125. This is read as “5 to the power of 3” or “5 cubed.”
When we study exponents, we essentially ask:
“How can we write repeated multiplication in a shorter, simpler way?”
Exponents allow us to work with very large and very small numbers efficiently.
Importance of Exponents
- Used in scientific notation (very large/small numbers like 3 × 10⁸ m/s)
- Essential for algebra, geometry, and calculus
- Helps simplify multiplication and division of repeated factors
- Used in compound interest, population growth, and radioactive decay
Example
5 × 5 × 5 × 5 = 5⁴ (5 to the fourth power) = 625
10 × 10 × 10 = 10³ (10 cubed) = 1000
2 × 2 × 2 × 2 × 2 = 2⁵ (2 to the fifth power) = 32
Subtopics
1. Base and Exponent
Base: The number being multiplied
Exponent (Power): The number that tells how many times the base is multiplied by itself
Exponential Form: base & exponent (e.g., 2⁴)
Expanded Form: base × base × base × … (repeated as many times as the exponent)
Example: 3⁵ = 3 × 3 × 3 × 3 × 3 = 243
2. Reading Exponents
|
Exponent |
Read As |
Meaning |
|
2 |
squared |
base × base |
|
3 |
cubed |
base × base × base |
|
4 |
to the fourth power |
base multiplied 4 times |
|
5 |
to the fifth power |
base multiplied 5 times |
Source: Kapdec.com
Examples:
7² = 49 (7 squared)
4³ = 64 (4 cubed)
2⁵ = 32 (2 to the fifth power)
3. Special Exponents
Exponent 1: Any number raised to the power 1 equals the number itself.
Example: 9¹ = 9, 100¹ = 100
Exponent 0: Any non-zero number raised to the power 0 equals 1.
Example: 5⁰ = 1, 100⁰ = 1, 1,000,000⁰ = 1
Note: 0⁰ is undefined (not covered in Grade 8).
4. First Law of Exponents – Product Rule
When multiplying two powers with the same base, add the exponents.
Formula: a^m × a^n = a^(m + n)
Example 1: 2³ × 2⁴ = 2^(3+4) = 2⁷ = 128
Check: 8 × 16 = 128 ✓
Example 2: 5² × 5³ = 5^(2+3) = 5⁵ = 3125
Example 3: x⁴ × x⁵ = x^(4+5) = x⁹
5. Second Law of Exponents – Quotient Rule
When dividing two powers with the same base, subtract the exponents.
Formula: a^m ÷ a^n = a^(m – n) (where m ≥ n, a ≠ 0)
Example 1: 2⁵ ÷ 2² = 2^(5-2) = 2³ = 8
Check: 32 ÷ 4 = 8 ✓
Example 2: 7⁶ ÷ 7³ = 7^(6-3) = 7³ = 343
Example 3: x⁸ ÷ x³ = x^(8-3) = x⁵
6. Third Law of Exponents – Power of a Power
When raising a power to another power, multiply the exponents.
Formula: (a^m)^n = a^(m × n)
Example 1: (2³)⁴ = 2^(3×4) = 2¹² = 4096
Check: (8)⁴ = 4096 ✓
Example 2: (5²)³ = 5^(2×3) = 5⁶ = 15625
Example 3: (x⁴)⁵ = x^(4×5) = x²⁰
7. Fourth Law of Exponents – Power of a Product
When raising a product to a power, raise each factor to that power.
Formula: (ab)^m = a^m × b^m
Example 1: (2 × 3)⁴ = 2⁴ × 3⁴ = 16 × 81 = 1296
Check: (6)⁴ = 1296 ✓
Example 2: (5x)³ = 5³ × x³ = 125x³
Example 3: (2y)⁵ = 2⁵ × y⁵ = 32y⁵
8. Fifth Law of Exponents – Power of a Quotient
When raising a quotient to a power, raise both numerator and denominator to that power.
Formula: (a/b)^m = a^m / b^m (b ≠ 0)
Example 1: (3/4)³ = 3³ / 4³ = 27/64
Example 2: (2x/5)² = (2x)² / 5² = 4x² / 25
9. Negative Exponents
A negative exponent means take the reciprocal of the base raised to the positive exponent.
Formula: a^(-n) = 1/a^n (a ≠ 0)
Example 1: 2^(-3) = 1/2³ = 1/8
Example 2: 5^(-2) = 1/5² = 1/25
Example 3: x^(-4) = 1/x⁴
Example 4: (2/3)^(-2) = (3/2)² = 9/4
Solved Examples
Example 1 – Product Rule: Simplify 3⁵ × 3²
Solution: 3^(5+2) = 3⁷ = 2187
Answer: 3⁷ (or 2187)
Example 2 – Quotient Rule: Simplify 8⁷ ÷ 8³
Solution: 8^(7-3) = 8⁴ = 4096
Answer: 8⁴ (or 4096)
Example 3 – Power of a Power: Simplify (4³)²
Solution: 4^(3×2) = 4⁶ = 4096
Answer: 4⁶ (or 4096)
Example 4 – Power of a Product: Simplify (3x)⁴
Solution: 3⁴ × x⁴ = 81x⁴
Answer: 81x⁴
Example 5 – Zero Exponent: Simplify 15⁰
Solution: Any non-zero number to the power 0 = 1
Answer: 1
Example 6 – Negative Exponent: Simplify 4^(-2)
Solution: 1/4² = 1/16
Answer: 1/16
Common Mistakes to Avoid
Mistake 1 – Adding exponents when multiplying different bases
2³ × 3² cannot be simplified using exponent laws (bases are different).
Correct understanding: Product rule only works when bases are the same.
Mistake 2 – Subtracting exponents incorrectly
5⁶ ÷ 5² = 5⁴, not 5³.
Correct understanding: 6 – 2 = 4, not 3.
Mistake 3 – Multiplying exponents when adding
(2³)⁴ = 2¹², not 2⁷.
Correct understanding: Power of a power multiplies exponents.
Mistake 4 – Forgetting the exponent applies to entire product
(3x)² = 9x², not 3x².
Correct understanding: Square both the coefficient AND the variable.
Mistake 5 – Thinking a⁰ = 0
Any non-zero number to the power 0 equals 1, not 0.
Correct understanding: 5⁰ = 1, 100⁰ = 1, a⁰ = 1 (a ≠ 0).
Mistake 6 – Misunderstanding negative exponents
2⁻³ = 1/8, not -8. The negative sign does NOT make the result negative.
Correct understanding: Negative exponent means reciprocal, not negative number.
Quick Reference Summary
Exponent Notation: a^m (a = base, m = exponent)
Product Rule: a^m × a^n = a^(m+n)
Quotient Rule: a^m ÷ a^n = a^(m-n) (a ≠ 0)
Power of a Power: (a^m)^n = a^(m×n)
Power of a Product: (ab)^m = a^m × b^m
Power of a Quotient: (a/b)^m = a^m / b^m (b ≠ 0)
Zero Exponent: a⁰ = 1 (a ≠ 0)
Negative Exponent: a^(-n) = 1/a^n (a ≠ 0)
Important: Laws apply only when bases are the same (for product/quotient rules)
