Problem Analysis:
The worksheet asks us to express various numbers with negative exponents as rational numbers in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). We will solve each question step by step using the properties of exponents.
#### Key Exponent Rules:
1.
Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)
2.
Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \)
Solution to Each Question:
####
Question 1: Expression of \( 2^{-3} \)
- Using the negative exponent rule: \( 2^{-3} = \frac{1}{2^3} \)
- Calculate \( 2^3 \): \( 2^3 = 8 \)
- Therefore, \( 2^{-3} = \frac{1}{8} \)
Answer: (B) \( \frac{1}{8} \)
####
Question 2: Expression of \( 3^{-3} \)
- Using the negative exponent rule: \( 3^{-3} = \frac{1}{3^3} \)
- Calculate \( 3^3 \): \( 3^3 = 27 \)
- Therefore, \( 3^{-3} = \frac{1}{27} \)
Answer: (D) \( \frac{1}{27} \)
####
Question 3: Expression of \( 4^{-3} \)
- Using the negative exponent rule: \( 4^{-3} = \frac{1}{4^3} \)
- Calculate \( 4^3 \): \( 4^3 = 64 \)
- Therefore, \( 4^{-3} = \frac{1}{64} \)
Answer: (B) \( \frac{1}{64} \)
####
Question 4: Expression of \( (-4)^{-2} \)
- Using the negative exponent rule: \( (-4)^{-2} = \frac{1}{(-4)^2} \)
- Calculate \( (-4)^2 \): \( (-4)^2 = 16 \) (since squaring a negative number results in a positive number)
- Therefore, \( (-4)^{-2} = \frac{1}{16} \)
Answer: (C) \( \frac{1}{16} \)
####
Question 5: Expression of \( (-2)^{-2} \)
- Using the negative exponent rule: \( (-2)^{-2} = \frac{1}{(-2)^2} \)
- Calculate \( (-2)^2 \): \( (-2)^2 = 4 \) (since squaring a negative number results in a positive number)
- Therefore, \( (-2)^{-2} = \frac{1}{4} \)
Answer: (C) \( \frac{1}{4} \)
Final Answers:
1. \( \boxed{\frac{1}{8}} \)
2. \( \boxed{\frac{1}{27}} \)
3. \( \boxed{\frac{1}{64}} \)
4. \( \boxed{\frac{1}{16}} \)
5. \( \boxed{\frac{1}{4}} \)
Parent Tip: Review the logic above to help your child master the concept of exponent worksheet 8th grade.
