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Work with exponents and powers
Learn step-by-step with practice exercises built right in.
What does it mean to raise a number to a power? Exponents are a shorthand way to show repeated multiplication - and they're everywhere in math, science, and the real world!
An exponent tells you how many times to multiply the base by itself.
Notation: bⁿ
Read as: "b to the nth power" or "b to the n"
Example: 2⁵
Powers show repeated multiplication:
2¹ = 2 (one factor of 2) 2² = 2 × 2 = 4 (two factors of 2) 2³ = 2 × 2 × 2 = 8 (three factors of 2) 2⁴ = 2 × 2 × 2 × 2 = 16 (four factors of 2) 2⁵ = 2 × 2 × 2 × 2 × 2 = 32 (five factors of 2)
Pattern: Each power is double the previous one!
Calculate 5³
Step 1: Identify the base and exponent. Base = 5 Exponent = 3
Step 2: Multiply the base by itself 3 times. 5³ = 5 × 5 × 5
Step 3: Calculate. 5 × 5 = 25 25 × 5 = 125
Answer: 5³ = 125
What is 10⁴?
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Squared (power of 2):
Cubed (power of 3):
Higher powers:
Example 1: Calculate 3⁴
3⁴ = 3 × 3 × 3 × 3 = 9 × 3 × 3 = 27 × 3 = 81
Answer: 3⁴ = 81
Example 2: Calculate 5³
5³ = 5 × 5 × 5 = 25 × 5 = 125
Answer: 5³ = 125
Example 3: Calculate 10⁴
10⁴ = 10 × 10 × 10 × 10 = 100 × 10 × 10 = 1,000 × 10 = 10,000
Answer: 10⁴ = 10,000
Powers of 10 are especially important!
10¹ = 10 10² = 100 10³ = 1,000 10⁴ = 10,000 10⁵ = 100,000 10⁶ = 1,000,000
Pattern: The exponent tells you how many zeros!
This is the basis for place value and scientific notation!
Any power of 1 equals 1:
1¹ = 1 1² = 1 × 1 = 1 1³ = 1 × 1 × 1 = 1 1¹⁰⁰ = 1
Why? Multiplying 1 by itself always gives 1!
Any number to the first power equals itself:
5¹ = 5 100¹ = 100 n¹ = n
Why? Using the number once means just the number!
Any non-zero number to the power of 0 equals 1:
5⁰ = 1 100⁰ = 1 n⁰ = 1 (where n ≠ 0)
This seems strange, but it's consistent with exponent rules you'll learn later!
Exception: 0⁰ is undefined (special case)
Be careful with negative numbers!
Even exponent: (-2)² = (-2) × (-2) = 4 (positive!) (-2)⁴ = (-2) × (-2) × (-2) × (-2) = 16 (positive!)
Odd exponent: (-2)³ = (-2) × (-2) × (-2) = -8 (negative!) (-2)⁵ = (-2) × (-2) × (-2) × (-2) × (-2) = -32 (negative!)
Rule:
With parentheses: (-3)² = (-3) × (-3) = 9
Without parentheses: -3² = -(3 × 3) = -9
BIG DIFFERENCE!
(-3)² means square the negative number -3² means find 3² then make it negative
Always use parentheses with negative bases!
Exponents come BEFORE multiplication and addition!
Example 1: 2 + 3² = 2 + 9 (exponent first!) = 11
Example 2: 2 × 3² = 2 × 9 = 18
Example 3: (2 + 3)² = 5² (parentheses first!) = 25
Remember: P-E-MDAS (Exponents are second!)
Which is larger: 2⁵ or 5²?
2⁵ = 32 5² = 25
So 2⁵ > 5²
Can't always tell just by looking - calculate if needed!
Example: Which is larger: 3⁴ or 4³? 3⁴ = 81 4³ = 64
So 3⁴ > 4³
Powers grow VERY fast!
Compare:
Just 10 doublings gets you over 1,000!
This is why exponential growth is so powerful (and sometimes dangerous, like with debt!)
Area of a square: A = s² Where s = side length
Volume of a cube: V = s³ Where s = side length
Surface area of a cube: SA = 6s²
Powers appear in many formulas!
Computer Science:
Population Growth:
Finance:
Geometry:
Physics:
Perfect Squares (n²): 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100 11² = 121, 12² = 144
Perfect Cubes (n³): 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1,000
Powers of 2: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32 2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2⁹ = 512, 2¹⁰ = 1,024
Powers of 10: 10¹ = 10, 10² = 100, 10³ = 1,000, 10⁴ = 10,000, etc.
Fractions can have exponents too!
Example: (1/2)³
(1/2)³ = (1/2) × (1/2) × (1/2) = 1/8
Rule: Raise numerator and denominator separately (a/b)ⁿ = aⁿ/bⁿ
Example: (2/3)² = 2²/3² = 4/9
Decimals with exponents:
Example: (0.5)² = 0.5 × 0.5 = 0.25
Example: (0.1)³ = 0.1 × 0.1 × 0.1 = 0.001
Notice: Powers of decimals less than 1 get smaller!
❌ Mistake 1: Multiplying instead of using power
❌ Mistake 2: Forgetting parentheses with negatives
❌ Mistake 3: Confusing exponent with multiplication
❌ Mistake 4: Wrong order of operations
❌ Mistake 5: Thinking 0⁰ = 1
To evaluate a power:
With order of operations:
With negative bases:
Most calculators have a power button:
Example: Calculate 7⁵
Check your calculator's manual for exact steps!
Key Rules:
Negative Bases:
Special Powers:
Order: Exponents before × ÷ + -
Tip 1: Memorize common powers
Tip 2: Write it out
Tip 3: Check with smaller examples
Tip 4: Watch for negative signs
Exponents show repeated multiplication:
Special cases:
Important rules:
Applications:
Mastering exponents is essential for algebra, science, and understanding how quantities grow and change!
Step 1: Recognize the pattern for powers of 10. 10⁴ means 1 followed by 4 zeros.
Step 2: Calculate. 10⁴ = 10 × 10 × 10 × 10 = 10,000
Shortcut: For 10ⁿ, write 1 followed by n zeros.
Answer: 10⁴ = 10,000
Simplify: 2³ × 2²
Step 1: Use the product rule for exponents. When multiplying same bases, ADD exponents. aᵐ × aⁿ = aᵐ⁺ⁿ
Step 2: Apply the rule. 2³ × 2² = 2³⁺² = 2⁵
Step 3: Calculate if needed. 2⁵ = 32
Answer: 2³ × 2² = 2⁵ = 32
Evaluate: (3²)³
Step 1: Use the power rule for exponents. When raising a power to a power, MULTIPLY exponents. (aᵐ)ⁿ = aᵐˣⁿ
Step 2: Apply the rule. (3²)³ = 3²ˣ³ = 3⁶
Step 3: Calculate. 3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 9 × 9 × 9 = 81 × 9 = 729
Answer: (3²)³ = 3⁶ = 729
A bacteria colony doubles every hour. If it starts with 5 bacteria, how many bacteria will there be after 6 hours? Express your answer using exponents, then calculate.
Step 1: Understand the pattern. Start: 5 bacteria After 1 hour: 5 × 2 = 10 After 2 hours: 5 × 2 × 2 = 5 × 2² After 3 hours: 5 × 2 × 2 × 2 = 5 × 2³
Step 2: Write the formula. After n hours: 5 × 2ⁿ
Step 3: Calculate for 6 hours. Bacteria = 5 × 2⁶
Step 4: Evaluate 2⁶. 2⁶ = 64
Step 5: Multiply. 5 × 64 = 320
Answer: 5 × 2⁶ = 320 bacteria after 6 hours