Worked examples: Applying multiple exponent rules.

📚 Quick Study Guide

• 🔢 Product of Powers: When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$
• ➗ Quotient of Powers: When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$
• 💪 Power of a Power: When raising a power to another power, multiply the exponents: $(a^m)^n = a^{m \cdot n}$
• 📦 Power of a Product: When raising a product to a power, distribute the exponent to each factor: $(ab)^n = a^n b^n$
• ➗ Power of a Quotient: When raising a quotient to a power, distribute the exponent to both the numerator and the denominator: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
• 0️⃣ Zero Exponent: Any nonzero number raised to the power of 0 is 1: $a^0 = 1$ (where $a \neq 0$)
• ➖ Negative Exponent: A negative exponent indicates a reciprocal: $a^{-n} = \frac{1}{a^n}$

🧪 Practice Quiz

1. What is the simplified form of $x^3 \cdot x^5$?
1. $x^2$
2. $x^8$
3. $x^{15}$
4. $2x^8$
2. Simplify: $\frac{y^7}{y^2}$
1. $y^9$
2. $y^5$
3. $y^{-5}$
4. $\frac{1}{y^5}$
3. Simplify: $(z^2)^4$
1. $z^6$
2. $z^8$
3. $4z^2$
4. $2z^4$
4. Simplify: $(2a^3)^2$
1. $2a^6$
2. $4a^5$
3. $4a^6$
4. $2a^5$
5. Simplify: $(\frac{b^4}{c^2})^3$
1. $\frac{b^7}{c^5}$
2. $\frac{b^{12}}{c^6}$
3. $\frac{b^1}{c^1}$
4. $\frac{b^{64}}{c^8}$
6. Simplify: $5^0 + x^0$ (assuming $x \neq 0$)
1. 0
2. 1
3. 2
4. 5
7. Simplify: $p^{-3}$
1. $-p^3$
2. $\frac{1}{p^3}$
3. $p^3$
4. $\frac{1}{-p^3}$