📊 Exponential Expressions

Essential Exponent Laws

• Product Rule: $a^m \cdot a^n = a^{m+n}$
  Example: $x^3 \cdot x^5 = x^8$
• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ (when $a \neq 0$)
  Example: $\frac{x^7}{x^3} = x^4$
• Power Rule: $(a^m)^n = a^{mn}$
  Example: $(x^2)^4 = x^8$
• Power of a Product: $(ab)^n = a^n b^n$
  Example: $(2x)^3 = 8x^3$
• Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
  Example: $\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}$

Special Cases

• $a^0 = 1$ (anything to the zero power equals 1, except $0^0$)
• $a^1 = a$ (anything to the first power equals itself)

The Rule

$$a^{-n} = \frac{1}{a^n}$$

Examples

• $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
• $x^{-2} = \frac{1}{x^2}$
• $5^{-1} = \frac{1}{5}$
• $\frac{1}{x^{-3}} = x^3$ (flip it!)

Working with Negative Exponents

Example: Simplify $\frac{x^{-3}y^4}{x^2y^{-1}}$

Step 1: Move negative exponents

$\frac{y^4 \cdot y^1}{x^3 \cdot x^2}$

Step 2: Apply product rule

$$\frac{y^5}{x^5}$$

The Connection

$$a^{\frac{1}{n}} = \sqrt[n]{a}$$

The denominator tells you which root to take!

Common Examples

• $16^{\frac{1}{2}} = \sqrt{16} = 4$
• $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$
• $27^{\frac{1}{3}} = \sqrt[3]{27} = 3$
• $81^{\frac{1}{4}} = \sqrt[4]{81} = 3$

More Complex Fractional Exponents

$$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

Example: Calculate $8^{\frac{2}{3}}$

Method 1: $\sqrt[3]{8^2} = \sqrt[3]{64} = 4$

Method 2: $(\sqrt[3]{8})^2 = 2^2 = 4$ ✨

Another Example: $27^{\frac{4}{3}}$

$= (\sqrt[3]{27})^4 = 3^4 = 81$

The Growth Formula

$$A(t) = A_0(1 + r)^t$$

• $A(t)$ = amount after time $t$
• $A_0$ = initial amount
• $r$ = growth rate (as a decimal)
• $t$ = time

Example 1: Population Growth

A town of 10,000 people grows at 3% per year. Find population after 5 years.

$A_0 = 10{,}000$, $r = 0.03$, $t = 5$

$A(5) = 10{,}000(1.03)^5$

$= 10{,}000(1.159) \approx 11{,}590$ people ✨

Example 2: Compound Interest

$\$1{,}000$ invested at 5% annual interest, compounded yearly, for 10 years:

$A(10) = 1{,}000(1.05)^{10} \approx \$1{,}628.89$ 💰

Doubling Time

How long to double? Solve $A_0(1 + r)^t = 2A_0$

This simplifies to $(1 + r)^t = 2$

The Decay Formula

$$A(t) = A_0(1 - r)^t$$

Or equivalently: $A(t) = A_0 \cdot b^t$ where $0 < b < 1$

Example 1: Radioactive Decay (Half-Life)

100g of a substance has a half-life of 5 years. How much remains after 15 years?

After each 5 years, half remains:

$A(15) = 100 \cdot \left(\frac{1}{2}\right)^{15/5} = 100 \cdot \left(\frac{1}{2}\right)^3$

$= 100 \cdot \frac{1}{8} = 12.5$ grams ☢️

Example 2: Car Depreciation

A car worth $\$25{,}000$ depreciates 15% per year:

$A(t) = 25{,}000(0.85)^t$

After 3 years: $A(3) = 25{,}000(0.85)^3 \approx \$15{,}344$ 🚗

Example 3: Cooling

Coffee at 180°F cools 10% every minute in a 70°F room:

Temperature above room temp: $(180 - 70)(0.9)^t + 70$