Exponents - Power Rule worksheet for simplifying expressions using the power rule, with problems labeled a-t.

The task involves simplifying expressions using the Power Rule of exponents. The Power Rule states:

$$
(a^m)^n = a^{m \cdot n}
$$

This means that when you have an exponent raised to another exponent, you multiply the exponents.

Additionally, we need to ensure that all answers are written with positive exponents. If a negative exponent appears in the final answer, we can rewrite it using the rule:

$$
a^{-n} = \frac{1}{a^n}
$$

Let's solve each problem step by step.

---

Problem 1: \( (s^t)^{-4} \)

Using the Power Rule:
$$
(s^t)^{-4} = s^{t \cdot (-4)} = s^{-4t}
$$
To write this with a positive exponent:
$$
s^{-4t} = \frac{1}{s^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{s^{4t}}}
$$

---

Problem 2: \( (v^t)^{-4} \)

Using the Power Rule:
$$
(v^t)^{-4} = v^{t \cdot (-4)} = v^{-4t}
$$
To write this with a positive exponent:
$$
v^{-4t} = \frac{1}{v^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{v^{4t}}}
$$

---

Problem 3: \( (k^t)^{-4} \)

Using the Power Rule:
$$
(k^t)^{-4} = k^{t \cdot (-4)} = k^{-4t}
$$
To write this with a positive exponent:
$$
k^{-4t} = \frac{1}{k^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{k^{4t}}}
$$

---

Problem 4: \( (o^t)^{-4} \)

Using the Power Rule:
$$
(o^t)^{-4} = o^{t \cdot (-4)} = o^{-4t}
$$
To write this with a positive exponent:
$$
o^{-4t} = \frac{1}{o^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{o^{4t}}}
$$

---

Problem 5: \( (u^t)^{-4} \)

Using the Power Rule:
$$
(u^t)^{-4} = u^{t \cdot (-4)} = u^{-4t}
$$
To write this with a positive exponent:
$$
u^{-4t} = \frac{1}{u^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{u^{4t}}}
$$

---

Problem 6: \( (z^t)^{-4} \)

Using the Power Rule:
$$
(z^t)^{-4} = z^{t \cdot (-4)} = z^{-4t}
$$
To write this with a positive exponent:
$$
z^{-4t} = \frac{1}{z^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{z^{4t}}}
$$

---

Problem 7: \( (C^{-t})^4 \)

Using the Power Rule:
$$
(C^{-t})^4 = C^{-t \cdot 4} = C^{-4t}
$$
To write this with a positive exponent:
$$
C^{-4t} = \frac{1}{C^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{C^{4t}}}
$$

---

Problem 8: \( (q^{-t})^4 \)

Using the Power Rule:
$$
(q^{-t})^4 = q^{-t \cdot 4} = q^{-4t}
$$
To write this with a positive exponent:
$$
q^{-4t} = \frac{1}{q^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{q^{4t}}}
$$

---

Problem 9: \( (n^{-t})^4 \)

Using the Power Rule:
$$
(n^{-t})^4 = n^{-t \cdot 4} = n^{-4t}
$$
To write this with a positive exponent:
$$
n^{-4t} = \frac{1}{n^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{n^{4t}}}
$$

---

Problem 10: \( (C^{-t})^4 \)

Using the Power Rule:
$$
(C^{-t})^4 = C^{-t \cdot 4} = C^{-4t}
$$
To write this with a positive exponent:
$$
C^{-4t} = \frac{1}{C^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{C^{4t}}}
$$

---

Problem 11: \( (D^{-t})^{-4} \)

Using the Power Rule:
$$
(D^{-t})^{-4} = D^{-t \cdot (-4)} = D^{4t}
$$
This is already in positive exponent form.
Final Answer:
$$
\boxed{D^{4t}}
$$

---

Problem 12: \( (t^{-t})^4 \)

Using the Power Rule:
$$
(t^{-t})^4 = t^{-t \cdot 4} = t^{-4t}
$$
To write this with a positive exponent:
$$
t^{-4t} = \frac{1}{t^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{t^{4t}}}
$$

---

Problem 13: \( (u^{-t})^4 \)

Using the Power Rule:
$$
(u^{-t})^4 = u^{-t \cdot 4} = u^{-4t}
$$
To write this with a positive exponent:
$$
u^{-4t} = \frac{1}{u^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{u^{4t}}}
$$

---

Problem 14: \( (y^{-t})^4 \)

Using the Power Rule:
$$
(y^{-t})^4 = y^{-t \cdot 4} = y^{-4t}
$$
To write this with a positive exponent:
$$
y^{-4t} = \frac{1}{y^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{y^{4t}}}
$$

---

Problem 15: \( (l^{-t})^4 \)

Using the Power Rule:
$$
(l^{-t})^4 = l^{-t \cdot 4} = l^{-4t}
$$
To write this with a positive exponent:
$$
l^{-4t} = \frac{1}{l^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{l^{4t}}}
$$

---

Problem 16: \( (t^{-t})^4 \)

Using the Power Rule:
$$
(t^{-t})^4 = t^{-t \cdot 4} = t^{-4t}
$$
To write this with a positive exponent:
$$
t^{-4t} = \frac{1}{t^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{t^{4t}}}
$$

---

Problem 17: \( (y^{-t})^4 \)

Using the Power Rule:
$$
(y^{-t})^4 = y^{-t \cdot 4} = y^{-4t}
$$
To write this with a positive exponent:
$$
y^{-4t} = \frac{1}{y^{4t}}
$$
Final Answer:
$$
\boxed{\frac{1}{y^{4t}}}
$$

---

Summary of Final Answers:

1. \( \boxed{\frac{1}{s^{4t}}} \)
2. \( \boxed{\frac{1}{v^{4t}}} \)
3. \( \boxed{\frac{1}{k^{4t}}} \)
4. \( \boxed{\frac{1}{o^{4t}}} \)
5. \( \boxed{\frac{1}{u^{4t}}} \)
6. \( \boxed{\frac{1}{z^{4t}}} \)
7. \( \boxed{\frac{1}{C^{4t}}} \)
8. \( \boxed{\frac{1}{q^{4t}}} \)
9. \( \boxed{\frac{1}{n^{4t}}} \)
10. \( \boxed{\frac{1}{C^{4t}}} \)
11. \( \boxed{D^{4t}} \)
12. \( \boxed{\frac{1}{t^{4t}}} \)
13. \( \boxed{\frac{1}{u^{4t}}} \)
14. \( \boxed{\frac{1}{y^{4t}}} \)
15. \( \boxed{\frac{1}{l^{4t}}} \)
16. \( \boxed{\frac{1}{t^{4t}}} \)
17. \( \boxed{\frac{1}{y^{4t}}} \)

---

Final Boxed Answers:
$$
\boxed{
\begin{aligned}
&1. \frac{1}{s^{4t}}, \quad 2. \frac{1}{v^{4t}}, \quad 3. \frac{1}{k^{4t}}, \quad 4. \frac{1}{o^{4t}}, \quad 5. \frac{1}{u^{4t}}, \quad 6. \frac{1}{z^{4t}}, \\
&7. \frac{1}{C^{4t}}, \quad 8. \frac{1}{q^{4t}}, \quad 9. \frac{1}{n^{4t}}, \quad 10. \frac{1}{C^{4t}}, \quad 11. D^{4t}, \\
&12. \frac{1}{t^{4t}}, \quad 13. \frac{1}{u^{4t}}, \quad 14. \frac{1}{y^{4t}}, \quad 15. \frac{1}{l^{4t}}, \quad 16. \frac{1}{t^{4t}}, \quad 17. \frac{1}{y^{4t}}
\end{aligned}
}
$$