Practice worksheet from Glencoe Pre-Algebra Lesson 4-7 focusing on negative exponents.

Problem Overview:

The task involves rewriting expressions using positive exponents and evaluating expressions given specific values. The goal is to practice working with negative exponents and their properties.

---

Step-by-Step Solution:

#### Part 1: Write each expression using a positive exponent

1. Expression: \(7^{-8}\)
- Using the property \(a^{-n} = \frac{1}{a^n}\):
\[
7^{-8} = \frac{1}{7^8}
\]

2. Expression: \((-4)^{-2}\)
- Using the property \(a^{-n} = \frac{1}{a^n}\):
\[
(-4)^{-2} = \frac{1}{(-4)^2}
\]

3. Expression: \((-1)^{-12}\)
- Using the property \(a^{-n} = \frac{1}{a^n}\):
\[
(-1)^{-12} = \frac{1}{(-1)^{12}}
\]
- Note: \((-1)^{12} = 1\) (since the exponent is even):
\[
(-1)^{-12} = \frac{1}{1} = 1
\]

4. Expression: \(c^{-4}\)
- Using the property \(a^{-n} = \frac{1}{a^n}\):
\[
c^{-4} = \frac{1}{c^4}
\]

5. Expression: \(g^{17}\)
- This expression already has a positive exponent, so no change is needed:
\[
g^{17} = g^{17}
\]

6. Expression: \(5x^{-4}\)
- Using the property \(a^{-n} = \frac{1}{a^n}\):
\[
5x^{-4} = 5 \cdot \frac{1}{x^4} = \frac{5}{x^4}
\]

---

#### Part 2: Write each fraction as an expression using a negative exponent

7. Fraction: \(\frac{1}{y^9}\)
- Using the property \(\frac{1}{a^n} = a^{-n}\):
\[
\frac{1}{y^9} = y^{-9}
\]

8. Fraction: \(\frac{1}{z^7}\)
- Using the property \(\frac{1}{a^n} = a^{-n}\):
\[
\frac{1}{z^7} = z^{-7}
\]

9. Fraction: \(\frac{1}{t^3}\)
- Using the property \(\frac{1}{a^n} = a^{-n}\):
\[
\frac{1}{t^3} = t^{-3}
\]

10. Fraction: \(\frac{1}{b^4}\)
- Using the property \(\frac{1}{a^n} = a^{-n}\):
\[
\frac{1}{b^4} = b^{-4}
\]

11. Fraction: \(\frac{1}{m^2}\)
- Using the property \(\frac{1}{a^n} = a^{-n}\):
\[
\frac{1}{m^2} = m^{-2}
\]

12. Fraction: \(\frac{1}{r^5}\)
- Using the property \(\frac{1}{a^n} = a^{-n}\):
\[
\frac{1}{r^5} = r^{-5}
\]

13. Fraction: \(\frac{1}{s^6}\)
- Using the property \(\frac{1}{a^n} = a^{-n}\):
\[
\frac{1}{s^6} = s^{-6}
\]

14. Fraction: \(\frac{1}{344}\)
- This can be written as:
\[
\frac{1}{344} = 344^{-1}
\]

---

#### Part 3: Evaluate each expression if \(x = 3\), \(y = -2\), and \(z = 6\)

15. Expression: \(x^{-4}\)
- Substitute \(x = 3\):
\[
x^{-4} = 3^{-4} = \frac{1}{3^4} = \frac{1}{81}
\]

16. Expression: \(y^{-2}\)
- Substitute \(y = -2\):
\[
y^{-2} = (-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4}
\]

17. Expression: \(z^{-1}\)
- Substitute \(z = 6\):
\[
z^{-1} = 6^{-1} = \frac{1}{6}
\]

18. Expression: \(x^4\)
- Substitute \(x = 3\):
\[
x^4 = 3^4 = 81
\]

19. Expression: \(y^2\)
- Substitute \(y = -2\):
\[
y^2 = (-2)^2 = 4
\]

20. Expression: \(z^2\)
- Substitute \(z = 6\):
\[
z^2 = 6^2 = 36
\]

---

#### Part 4: Real-world application

21. Problem: Hair grows at a rate of \(\frac{1}{25}\) inch per day. Write this number using negative exponents.
- The given rate is \(\frac{1}{25}\). Using the property \(\frac{1}{a} = a^{-1}\):
\[
\frac{1}{25} = 25^{-1}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
&1. \frac{1}{7^8}, \quad 2. \frac{1}{(-4)^2}, \quad 3. 1, \quad 4. \frac{1}{c^4}, \quad 5. g^{17}, \quad 6. \frac{5}{x^4}, \\
&7. y^{-9}, \quad 8. z^{-7}, \quad 9. t^{-3}, \quad 10. b^{-4}, \quad 11. m^{-2}, \quad 12. r^{-5}, \quad 13. s^{-6}, \quad 14. 344^{-1}, \\
&15. \frac{1}{81}, \quad 16. \frac{1}{4}, \quad 17. \frac{1}{6}, \quad 18. 81, \quad 19. 4, \quad 20. 36, \quad 21. 25^{-1}.
\end{aligned}
}
\]