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Basic Exponents

1. What is an Exponent?

An exponent tells us how many times a number (called the base) is multiplied by itself.

Example:

23 = 2 × 2 × 2 = 8
Base = 2 (the number being multiplied)
Exponent = 3 (the number of times it's multiplied)

2. Understanding Exponential Notation

Instead of writing repeated multiplication, we use exponents.

Example 1: 52 = 5 × 5 = 25

Example 2: 34 = 3 × 3 × 3 × 3 = 81

Example 3: 101 = 10

Key Idea: The exponent tells how many times to multiply the base by itself.

3. Special Exponent Rules

Rule 1: Any Number to the Power of 1
Any number raised to the power of 1 is itself: a1 = a
Example: 71 = 7 and (-3)1 = -3

Rule 2: Any Number to the Power of 0
Any number (except 0) raised to the power of 0 is 1: a0 = 1
Example: 100 = 1, (-5)0 = 1, 10000 = 1

Rule 3: Multiplying Powers with the Same Base
am × an = am+n
Example: 23 × 24 = 27 = 128 and x5 × x2 = x7

Rule 4: Dividing Powers with the Same Base
am ÷ an = am-n
Example: 56 ÷ 52 = 54 = 625 and x9 ÷ x3 = x6

Rule 5: Power of a Power Rule
(am)n = am×n
Example: (32)3 = 36 = 729 and (x4)2 = x8

Rule 6: Power of a Product Rule
(ab)n = an × bn
Example: (2×5)3 = 23 × 53 = 8 × 125 = 1000

Rule 7: Power of a Fraction Rule
When raising a fraction to an exponent, apply the exponent to both the numerator and denominator: (a/b)n = an ÷ bn
Example: (3/4)2 = 9/16

4. Real-Life Applications of Exponents

  • Computer Memory & Storage: Data is stored in powers of 2 (e.g., kilobytes, megabytes, gigabytes).
  • Science & Physics: Formulas like the area of a circle, πr2, use exponents.
  • Finance & Interest: Compound interest follows formulas like A = P(1+r)t.

Practice Questions 🎯

  1. Solve: 43
  2. Rewrite using exponents: 5 × 5 × 5 × 5
  3. What is 20?
  4. Solve: (32)3
  5. Simplify: x7 × x3
  6. Simplify: 105 ÷ 102
  7. Expand: (2y)4
  8. Convert to fraction: (1/3)2
  9. Solve: 62 × 63
  10. Simplify: (a3b2)2

Want a Bigger Challenge? 🤔🔥

  1. Evaluate: (23)4 ÷ 25
  2. Simplify: x10y5 ÷ x3y2
  3. Solve: (43 × 42) ÷ 44
  4. If a bacteria culture doubles every hour, how many bacteria are there after 6 hours if it starts with 10?
  5. The population of a city grows exponentially. If the population starts at 5000 and doubles every 10 years, how many people will there be in 30 years?

7. Summary

  • ✅ Exponents represent repeated multiplication.
  • ✅ Key exponent rules:
    • am × an = am+n
    • am ÷ an = am-n
    • (am)n = am×n
    • a0 = 1
  • ✅ Exponents are widely used in science, computing, and finance.

🎉 Awesome work! Now that you understand Basic Exponents, you're ready for the next topic: Introduction to Functions! 🚀

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