Exponents and Powers

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!

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What Is an Exponent?

An exponent tells you how many times to multiply the base by itself.

Notation: bⁿ

• b = base (the number being multiplied)
• n = exponent or power (how many times)

Read as: "b to the nth power" or "b to the n"

Example: 2⁵

• Base: 2
• Exponent: 5
• Meaning: 2 × 2 × 2 × 2 × 2 = 32

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Understanding Powers

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!

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Special Names for Powers

Squared (power of 2):

• 5² = "five squared" = 5 × 5 = 25
• Called "squared" because it's the area of a square

Cubed (power of 3):

• 4³ = "four cubed" = 4 × 4 × 4 = 64
• Called "cubed" because it's the volume of a cube

Higher powers:

• n⁴ = "n to the fourth power"
• n⁵ = "n to the fifth power"
• And so on...

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Evaluating 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

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Powers of 10

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!

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Base of 1

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!

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Exponent of 1

Any number to the first power equals itself:

5¹ = 5 100¹ = 100 n¹ = n

Why? Using the number once means just the number!

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Exponent of 0

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)

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Negative Bases

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:

• Even exponent → Positive result
• Odd exponent → Negative result

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Parentheses Matter!

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!

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Order of Operations (PEMDAS)

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!)

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Comparing Powers

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

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Exponential Growth

Powers grow VERY fast!

Compare:

• 2¹ = 2
• 2² = 4
• 2³ = 8
• 2⁴ = 16
• 2⁵ = 32
• 2⁶ = 64
• 2⁷ = 128
• 2⁸ = 256
• 2⁹ = 512
• 2¹⁰ = 1,024

Just 10 doublings gets you over 1,000!

This is why exponential growth is so powerful (and sometimes dangerous, like with debt!)

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Powers in Formulas

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!

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Real-World Applications

Computer Science:

• Data storage: 2¹⁰ = 1,024 bytes = 1 kilobyte
• Binary: 2⁸ = 256 possible values in a byte

Population Growth:

• Bacteria doubling: starts at 1, after n doublings = 2ⁿ

Finance:

• Compound interest uses exponents
• Money growing over time

Geometry:

• Areas use power of 2 (square units)
• Volumes use power of 3 (cubic units)

Physics:

• Distance: d = 16t² (falling objects)
• Energy: E = mc²

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Perfect Powers to Memorize

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.

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Fractional Bases

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

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Decimal Bases

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!

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Common Mistakes to Avoid

❌ Mistake 1: Multiplying instead of using power

• Wrong: 3⁴ = 3 × 4 = 12
• Right: 3⁴ = 3 × 3 × 3 × 3 = 81

❌ Mistake 2: Forgetting parentheses with negatives

• Wrong: -3² = 9
• Right: -3² = -9, but (-3)² = 9

❌ Mistake 3: Confusing exponent with multiplication

• 2³ ≠ 2 × 3
• 2³ = 8, not 6

❌ Mistake 4: Wrong order of operations

• Wrong: 2 + 3² = 5² = 25
• Right: 2 + 3² = 2 + 9 = 11

❌ Mistake 5: Thinking 0⁰ = 1

• 0⁰ is undefined (special case)
• But n⁰ = 1 for any n ≠ 0

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Problem-Solving Strategy

To evaluate a power:

1. Identify base and exponent
2. Write out repeated multiplication
3. Calculate step by step
4. Check: Does the answer make sense?

With order of operations:

1. Do what's in parentheses first
2. Then evaluate exponents
3. Then multiply/divide
4. Finally add/subtract

With negative bases:

1. Use parentheses!
2. Count factors
3. Even factors → positive
4. Odd factors → negative

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Using a Calculator

Most calculators have a power button:

• Often labeled: ^ or yx or xʸ
• Some: Use shift + another key

Example: Calculate 7⁵

• Enter: 7
• Press: ^ (or power key)
• Enter: 5
• Press: =
• Result: 16,807

Check your calculator's manual for exact steps!

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Quick Reference

Key Rules:

• bⁿ = b × b × ... × b (n times)
• b¹ = b
• b⁰ = 1 (if b ≠ 0)
• 1ⁿ = 1

Negative Bases:

• Even exponent → positive
• Odd exponent → negative
• Use parentheses!

Special Powers:

• n² = "squared"
• n³ = "cubed"
• 10ⁿ has n zeros

Order: Exponents before × ÷ + -

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Practice Tips

Tip 1: Memorize common powers

• Squares through 12²
• Cubes through 5³
• Powers of 2 through 2¹⁰
• Powers of 10

Tip 2: Write it out

• Don't try to do big powers in your head
• Write the repeated multiplication
• Calculate step by step

Tip 3: Check with smaller examples

• If confused, try with exponent 2 or 3
• Pattern usually becomes clear

Tip 4: Watch for negative signs

• Are parentheses there?
• Count the factors to check sign

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Summary

Exponents show repeated multiplication:

• bⁿ means multiply b by itself n times
• Base = number being multiplied
• Exponent = how many times

Special cases:

• b⁰ = 1
• b¹ = b
• 1ⁿ = 1
• n² = squared (area)
• n³ = cubed (volume)

Important rules:

• Negative base with even exponent → positive
• Negative base with odd exponent → negative
• Exponents in PEMDAS come after parentheses, before multiplication
• Powers grow very quickly!

Applications:

• Geometry (area, volume)
• Science (E = mc²)
• Finance (compound interest)
• Computer science (binary, data)
• Real-world growth

Mastering exponents is essential for algebra, science, and understanding how quantities grow and change!