Quick quiz: Evaluate expressions with negative exponents

📚 Quick Study Guide

• 🔢 Definition: A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. For example, $x^{-n} = \frac{1}{x^n}$.
• 💡 Rule 1: Any non-zero number raised to the power of -1 is equal to its reciprocal: $a^{-1} = \frac{1}{a}$.
• ➗ Rule 2: When dividing exponential terms with the same base, subtract the exponents: $\frac{x^a}{x^b} = x^{a-b}$. This can result in negative exponents if $b > a$.
• ➕ Rule 3: Remember the order of operations (PEMDAS/BODMAS). Exponents are evaluated before multiplication, division, addition, or subtraction.
• 📝 Important Note: Be careful with signs! A negative exponent does NOT make the base negative.
• 📌 Fractional Bases: If the base is a fraction, inverting the fraction and changing the sign of the exponent can simplify calculations: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.

Practice Quiz

1. What is the value of $2^{-3}$?
   1. $\frac{1}{8}$
   2. -8
   3. 6
   4. $\frac{1}{6}$
2. Simplify $x^{-5}$:
   1. $\frac{1}{x^5}$
   2. -5x
   3. $5x$
   4. $x^5$
3. What is the value of $(\frac{1}{3})^{-2}$?
   1. 9
   2. $\frac{1}{9}$
   3. -9
   4. $\frac{-1}{9}$
4. Simplify $4^{-2}$:
   1. $\frac{1}{16}$
   2. 16
   3. -8
   4. $\frac{1}{8}$
5. Evaluate $(-2)^{-4}$:
   1. $\frac{1}{16}$
   2. -16
   3. 16
   4. $\frac{-1}{16}$
6. Simplify $\frac{x^2}{x^5}$:
   1. $x^{-3}$
   2. $x^3$
   3. $\frac{1}{x^7}$
   4. $x^7$
7. What is the value of $10^{-1}$?
   1. $\frac{1}{10}$
   2. -10
   3. 10
   4. 1