Step-by-step examples of simplifying expressions with zero and negative exponents

📚 Quick Study Guide

• 🔢 Zero Exponent Rule: Any non-zero number raised to the power of 0 is equal to 1. Mathematically, for any $a \neq 0$, $a^0 = 1$.
• ➖ Negative Exponent Rule: Any non-zero number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent. Mathematically, for any $a \neq 0$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.
• ➗ Quotient of Powers Rule: When dividing exponential terms with the same base, subtract the exponents. Mathematically, $\frac{a^m}{a^n} = a^{m-n}$. This rule is especially useful when simplifying expressions with negative exponents.
• 💡 Tips for Simplifying: First, deal with negative exponents by moving the terms to the opposite side of the fraction (numerator to denominator or vice versa). Then, simplify using the quotient of powers rule.
• 📝 Important Note: Remember that these rules apply only when the base is non-zero. $0^0$ and $0^{-n}$ are undefined.

🧪 Practice Quiz

1. What is the simplified form of $5^0$?
1. A. 0
2. B. 1
3. C. 5
4. D. Undefined
2. What is the simplified form of $3^{-2}$?
1. A. -9
2. B. $\frac{1}{9}$
3. C. -6
4. D. 9
3. Simplify the expression: $\frac{x^5}{x^8}$
1. A. $x^{13}$
2. B. $x^{-3}$
3. C. $\frac{1}{x^3}$
4. D. Both B and C
4. Simplify: $(2a)^{-3}$
1. A. $\frac{1}{8a^3}$
2. B. $-6a$
3. C. $\frac{1}{2a^3}$
4. D. $\frac{1}{-8a^3}$
5. Simplify the expression: $4x^0 + 2y^{-1}$ (assuming y ≠ 0)
1. A. $6 + \frac{1}{y}$
2. B. $4 + \frac{2}{y}$
3. C. $\frac{6}{y}$
4. D. $6y$
6. What is the value of $(-10)^0$?
1. A. -10
2. B. 0
3. C. 1
4. D. -1
7. Simplify: $\frac{5a^{-2}}{b^{-3}}$
1. A. $\frac{5b^3}{a^2}$
2. B. $\frac{5}{a^2b^3}$
3. C. $\frac{5a^2}{b^3}$
4. D. $5a^2b^3$