Exponents - Sixth Grade
Complete Notes & Formulas
What are Exponents?
Exponents show how many times a number (called the base) is multiplied by itself. They are also called powers.
baseexponent = power
Example: 2³ = 2 × 2 × 2 = 8
Key Vocabulary
Base: The number being multiplied (e.g., in 5³, 5 is the base)
Exponent: The small number that tells how many times to multiply the base (e.g., in 5³, 3 is the exponent)
Power: The complete expression or the result (e.g., 5³ is a power; the value 125 is also the power)
Reading Exponents
• 5² = "five squared" or "five to the second power"
• 5³ = "five cubed" or "five to the third power"
• 5⁴ = "five to the fourth power"
• 5⁵ = "five to the fifth power"
1. Write Multiplication Expressions Using Exponents
Converting Repeated Multiplication to Exponents
a × a × a × a × ... (n times) = an
Example 1: Simple Conversion
Problem: Write 7 × 7 × 7 × 7 × 7 using exponents
Step 1: Identify the base (repeated number): 7
Step 2: Count how many times it's multiplied: 5 times
Step 3: Write as baseexponent
Answer: 7⁵
Example 2: Multiple Terms
Problem: Write 3 × 3 × 5 × 5 × 5 using exponents
3 appears 2 times → 3²
5 appears 3 times → 5³
Answer: 3² × 5³
Example 3: With Variables
Problem: Write x × x × x × y × y using exponents
x appears 3 times → x³
y appears 2 times → y²
Answer: x³y²
2. Evaluate Powers with Whole Number Bases
Steps to Evaluate Powers
Step 1: Identify the base and exponent
Step 2: Write the base multiplied by itself (exponent number of times)
Step 3: Calculate the result
Example 1: Evaluate 4³
Base = 4, Exponent = 3
4³ = 4 × 4 × 4
= 16 × 4
= 64
Answer: 64
Example 2: Evaluate 2⁵
2⁵ = 2 × 2 × 2 × 2 × 2
= 4 × 2 × 2 × 2
= 8 × 2 × 2
= 16 × 2
= 32
Answer: 32
Special Cases
Zero Exponent: a⁰ = 1 (for any non-zero a)
Example: 5⁰ = 1, 100⁰ = 1
One Exponent: a¹ = a
Example: 7¹ = 7, 25¹ = 25
3. Write Powers of Ten with Exponents
Understanding Powers of 10
Powers of 10 are special because they show place value. Each power of 10 adds one zero!
10n = 1 followed by n zeros
Powers of 10 Chart
| Exponential Form | Expanded Form | Standard Form |
|---|---|---|
| 10¹ | 10 | 10 |
| 10² | 10 × 10 | 100 |
| 10³ | 10 × 10 × 10 | 1,000 |
| 10⁴ | 10 × 10 × 10 × 10 | 10,000 |
| 10⁵ | 10 × 10 × 10 × 10 × 10 | 100,000 |
| 10⁶ | 10 × 10 × 10 × 10 × 10 × 10 | 1,000,000 |
Example: Write in Exponential Form
Problem: Write 1,000,000 using exponents
Count the zeros: 1,000,000 has 6 zeros
The exponent equals the number of zeros
Answer: 10⁶
4. Find the Missing Exponent or Base
Finding a Missing Exponent
Problem: Find the missing exponent: 3? = 81
Method: Keep multiplying the base until you reach the answer
3¹ = 3
3² = 9
3³ = 27
3⁴ = 81 ✓
Answer: Exponent is 4 (3⁴ = 81)
Finding a Missing Base
Problem: Find the missing base: ?³ = 125
Method: Think: What number multiplied 3 times equals 125?
Try: 2³ = 8 (too small)
Try: 3³ = 27 (too small)
Try: 4³ = 64 (too small)
Try: 5³ = 125 ✓
Answer: Base is 5 (5³ = 125)
Example: Powers of 10
Problem: 10? = 10,000
Count the zeros in 10,000: 4 zeros
The exponent equals the number of zeros
Answer: 10⁴ = 10,000
5. Evaluate Powers with Decimal Bases
Same Process, Decimal Base
When the base is a decimal, follow the same steps: multiply the base by itself (exponent times).
(decimal)exponent = decimal × decimal × ... (exponent times)
Example 1: Evaluate (0.5)²
(0.5)² = 0.5 × 0.5
= 0.25
Answer: 0.25
Example 2: Evaluate (0.2)³
(0.2)³ = 0.2 × 0.2 × 0.2
= 0.04 × 0.2
= 0.008
Answer: 0.008
Example 3: Evaluate (1.5)²
(1.5)² = 1.5 × 1.5
= 2.25
Answer: 2.25
6. Evaluate Powers with Fractional Bases
How to Evaluate Fraction Powers
(a/b)n = an/bn
Apply the exponent to BOTH numerator and denominator
Example 1: Evaluate (1/2)²
Method 1: Apply exponent to numerator and denominator
(1/2)² = 1²/2²
= 1/4
Method 2: Multiply the fraction by itself
(1/2)² = 1/2 × 1/2
= (1 × 1)/(2 × 2)
= 1/4
Answer: 1/4
Example 2: Evaluate (2/3)³
(2/3)³ = 2³/3³
= (2 × 2 × 2)/(3 × 3 × 3)
= 8/27
Answer: 8/27
Example 3: Evaluate (3/4)²
(3/4)² = 3²/4²
= 9/16
Answer: 9/16
7. Evaluate Powers (All Types)
Practice with Different Bases
| Expression | Type | Result |
|---|---|---|
| 6² | Whole number | 36 |
| (0.3)² | Decimal | 0.09 |
| (1/5)² | Fraction | 1/25 |
| 10³ | Power of 10 | 1,000 |
| (1.2)² | Decimal | 1.44 |
| (4/5)³ | Fraction | 64/125 |
Quick Reference: Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Basic Power | an = a × a × ... (n times) | 3³ = 27 |
| Zero Exponent | a⁰ = 1 | 5⁰ = 1 |
| One Exponent | a¹ = a | 7¹ = 7 |
| Fraction Base | (a/b)n = an/bn | (2/3)² = 4/9 |
| Powers of 10 | 10n = 1 + n zeros | 10⁴ = 10,000 |
💡 Important Tips to Remember
✓ The base is the number being multiplied
✓ The exponent tells how many times to multiply
✓ Any number to the 0 power = 1 (except 0⁰)
✓ Any number to the 1st power = itself
✓ Powers of 10: count the zeros
✓ For fractions: apply exponent to top AND bottom
✓ For decimals: multiply carefully, watch decimal places
✓ 2² = "two squared" (not "two two")
✓ 2³ = "two cubed" (not "two three")
✓ Check your work by multiplying it out
🧠 Memory Tricks & Strategies
What is an Exponent?
"The little number up high tells you how many times to multiply!"
Base vs. Exponent:
BASE is big (the main number)
EXPONENT is elevated (up high)
Powers of 10:
"Count the zeros, that's your power!"
1,000 has 3 zeros → 10³
Squared and Cubed:
Squared (²) = area of a SQUARE (2 dimensions)
Cubed (³) = volume of a CUBE (3 dimensions)
Fractions with Exponents:
"Top and bottom both get the power!"
(2/3)² = 2²/3² = 4/9
Zero Exponent Rule:
"Anything to the zero = hero (1)!"
Master Exponents! ⁿ 📐 ²³
Practice with different bases - whole numbers, decimals, and fractions!