Multiplying Exponents (A) worksheet with 10 problems to simplify expressions using negative exponents.

To solve the given problems involving multiplying exponents, we will use the following key properties of exponents:

1. Product of Powers Property: When multiplying powers with the same base, add the exponents:
\[
a^m \cdot a^n = a^{m+n}
\]

2. Negative Exponent Rule: A negative exponent means taking the reciprocal of the base:
\[
a^{-n} = \frac{1}{a^n}
\]

3. Zero Exponent Rule: Any non-zero number raised to the power of 0 is 1:
\[
a^0 = 1
\]

Let's solve each problem step by step.

---

Problem 1: \( 4^{-2} \cdot 4^{-1} \)

Using the Product of Powers Property:
\[
4^{-2} \cdot 4^{-1} = 4^{-2 + (-1)} = 4^{-3}
\]

Using the Negative Exponent Rule:
\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}
\]

Answer:
\[
\boxed{\frac{1}{64}}
\]

---

Problem 2: \( (-9)^5 \cdot (-9)^{-5} \)

Using the Product of Powers Property:
\[
(-9)^5 \cdot (-9)^{-5} = (-9)^{5 + (-5)} = (-9)^0
\]

Using the Zero Exponent Rule:
\[
(-9)^0 = 1
\]

Answer:
\[
\boxed{1}
\]

---

Problem 3: \( 2^{-1} \cdot 2^{-7} \)

Using the Product of Powers Property:
\[
2^{-1} \cdot 2^{-7} = 2^{-1 + (-7)} = 2^{-8}
\]

Using the Negative Exponent Rule:
\[
2^{-8} = \frac{1}{2^8} = \frac{1}{256}
\]

Answer:
\[
\boxed{\frac{1}{256}}
\]

---

Problem 4: \( (-5)^6 \cdot (-5)^{-2} \)

Using the Product of Powers Property:
\[
(-5)^6 \cdot (-5)^{-2} = (-5)^{6 + (-2)} = (-5)^4
\]

Since the base is \(-5\) and the exponent is even (4), the result is positive:
\[
(-5)^4 = 5^4 = 625
\]

Answer:
\[
\boxed{625}
\]

---

Problem 5: \( (-3)^5 \cdot (-3)^4 \)

Using the Product of Powers Property:
\[
(-3)^5 \cdot (-3)^4 = (-3)^{5 + 4} = (-3)^9
\]

Since the base is \(-3\) and the exponent is odd (9), the result is negative:
\[
(-3)^9 = -3^9 = -19683
\]

Answer:
\[
\boxed{-19683}
\]

---

Problem 6: \( (-8)^{-1} \cdot (-8)^5 \)

Using the Product of Powers Property:
\[
(-8)^{-1} \cdot (-8)^5 = (-8)^{-1 + 5} = (-8)^4
\]

Since the base is \(-8\) and the exponent is even (4), the result is positive:
\[
(-8)^4 = 8^4 = 4096
\]

Answer:
\[
\boxed{4096}
\]

---

Problem 7: \( 7^1 \cdot 7^8 \)

Using the Product of Powers Property:
\[
7^1 \cdot 7^8 = 7^{1 + 8} = 7^9
\]

Answer:
\[
\boxed{7^9}
\]

---

Problem 8: \( 7^0 \cdot 7^{-9} \)

Using the Zero Exponent Rule:
\[
7^0 = 1
\]

So:
\[
7^0 \cdot 7^{-9} = 1 \cdot 7^{-9} = 7^{-9}
\]

Using the Negative Exponent Rule:
\[
7^{-9} = \frac{1}{7^9}
\]

Answer:
\[
\boxed{\frac{1}{7^9}}
\]

---

Problem 9: \( 4^{-5} \cdot 4^2 \)

Using the Product of Powers Property:
\[
4^{-5} \cdot 4^2 = 4^{-5 + 2} = 4^{-3}
\]

Using the Negative Exponent Rule:
\[
4^{-3} = \frac{1}{4^3} = \frac{1}{64}
\]

Answer:
\[
\boxed{\frac{1}{64}}
\]

---

Problem 10: \( (-4)^{-4} \cdot (-4)^{-1} \)

Using the Product of Powers Property:
\[
(-4)^{-4} \cdot (-4)^{-1} = (-4)^{-4 + (-1)} = (-4)^{-5}
\]

Using the Negative Exponent Rule:
\[
(-4)^{-5} = \frac{1}{(-4)^5}
\]

Since the base is \(-4\) and the exponent is odd (5), the result is negative:
\[
(-4)^5 = -4^5 = -1024
\]

Thus:
\[
(-4)^{-5} = \frac{1}{-1024} = -\frac{1}{1024}
\]

Answer:
\[
\boxed{-\frac{1}{1024}}
\]

---

Final Answers:

1. \(\boxed{\frac{1}{64}}\)
2. \(\boxed{1}\)
3. \(\boxed{\frac{1}{256}}\)
4. \(\boxed{625}\)
5. \(\boxed{-19683}\)
6. \(\boxed{4096}\)
7. \(\boxed{7^9}\)
8. \(\boxed{\frac{1}{7^9}}\)
9. \(\boxed{\frac{1}{64}}\)
10. \(\boxed{-\frac{1}{1024}}\)