💪 Basic Exponents
Discover the power of exponents! Learn how to work with repeated multiplication and unlock exponential thinking.
⚡ What are Exponents?
An exponent tells you how many times to multiply a number (the base) by itself!
Notation and Terminology
In $3^4$:
- $3$ is the base (the number being multiplied)
- $4$ is the exponent or power (how many times)
- Read as: "3 to the 4th power" or "3 to the power of 4"
$$3^4 = 3 \times 3 \times 3 \times 3 = 81$$
🔢 Special Exponents
Some exponents have special names and meanings!
Important Cases
- Squared ($x^2$): Second power
$5^2 = 5 \times 5 = 25$ - Cubed ($x^3$): Third power
$4^3 = 4 \times 4 \times 4 = 64$ - Power of 1 ($x^1$): Always equals the base
$7^1 = 7$ - Power of 0 ($x^0$): Always equals 1 (for $x \neq 0$)
$5^0 = 1$, $100^0 = 1$ ✨
✖️ Product Rule
When multiplying powers with the same base, add the exponents!
Rule: $a^m \times a^n = a^{m+n}$
Example 1:
$$2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$$
Example 2:
$$x^2 \times x^5 = x^7$$
💡 Why? $(x \times x) \times (x \times x \times x \times x \times x) = x^7$
➗ Quotient Rule
When dividing powers with the same base, subtract the exponents!
Rule: $\frac{a^m}{a^n} = a^{m-n}$
Example 1:
$$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$$
Example 2:
$$\frac{x^8}{x^3} = x^5$$
🎯 Power Rule
When raising a power to another power, multiply the exponents!
Rule: $(a^m)^n = a^{m \times n}$
Example 1:
$$(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$$
Example 2:
$$(x^4)^2 = x^8$$
🌟 Real-Life Applications
- 💻 Computer Storage: $2^{10}$ bytes = 1 kilobyte (1024 bytes)
- 🦠 Biology: Bacteria double: 1 → 2 → 4 → 8 (powers of 2!)
- 💰 Compound Interest: Money grows exponentially over time
- 📱 Social Media: Viral content spreads exponentially