Worksheet featuring a cartoon eye with a magnifying glass, illustrating the concept of simplifying expressions with negative exponents.

Absolutely! Let’s solve each problem step by step, applying the negative exponent rule:

> Negative Exponent Rule:
> For any nonzero number \( a \) and integer \( n \),
> \[
a^{-n} = \frac{1}{a^n} \quad \text{or} \quad \frac{1}{a^{-n}} = a^n
\]

Also, we’ll use:
- Power of a power: \( (a^m)^n = a^{m \cdot n} \)
- Product of powers: \( a^m \cdot a^n = a^{m+n} \)
- Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \)
- Power of a product/quotient: \( (ab)^n = a^n b^n \), \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)

---

Problem 1: \( \left( \frac{-6}{x} \right)^{-3} \)

Apply the negative exponent rule to the entire fraction:

\[
\left( \frac{-6}{x} \right)^{-3} = \left( \frac{x}{-6} \right)^3 = \frac{x^3}{(-6)^3} = \frac{x^3}{-216} = -\frac{x^3}{216}
\]

✔ Answer: \( \boxed{-\dfrac{x^3}{216}} \)

---

Problem 2: \( \left( \frac{2x^{-2}}{y^{-3}} \right)^4 \)

First, simplify inside the parentheses using negative exponents:

\[
\frac{2x^{-2}}{y^{-3}} = 2 \cdot x^{-2} \cdot y^{3} = \frac{2y^3}{x^2}
\]

Now raise to the 4th power:

\[
\left( \frac{2y^3}{x^2} \right)^4 = \frac{(2)^4 (y^3)^4}{(x^2)^4} = \frac{16 y^{12}}{x^8}
\]

✔ Answer: \( \boxed{\dfrac{16y^{12}}{x^8}} \)

---

Problem 3: \( (18 + a)^{-2} \)

This is a binomial raised to a negative exponent. Apply the rule directly:

\[
(18 + a)^{-2} = \frac{1}{(18 + a)^2}
\]

We can leave it like this unless asked to expand the denominator.

✔ Answer: \( \boxed{\dfrac{1}{(18 + a)^2}} \)

---

Problem 4: \( \left( \frac{2^{-6}}{(x+2)^2} \right)^{-1} \)

Apply the outer exponent \(-1\) to both numerator and denominator:

\[
= \frac{(2^{-6})^{-1}}{((x+2)^2)^{-1}} = \frac{2^{6}}{(x+2)^{-2}} = 2^6 \cdot (x+2)^2 = 64(x+2)^2
\]

✔ Answer: \( \boxed{64(x+2)^2} \)

---

Problem 5: \( \left( s t^{-3} \cdot \frac{1}{s^{-4}} \right)^2 \)

First, simplify inside the parentheses:

Note: \( \frac{1}{s^{-4}} = s^4 \)

So,

\[
s t^{-3} \cdot s^4 = s^{1+4} \cdot t^{-3} = s^5 t^{-3}
\]

Now square the result:

\[
(s^5 t^{-3})^2 = s^{10} t^{-6} = \frac{s^{10}}{t^6}
\]

✔ Answer: \( \boxed{\dfrac{s^{10}}{t^6}} \)

---

Problem 6: \( -\left( \frac{x y^{-2}}{4x} \right)^{-2} \)

First, simplify inside the parentheses:

\[
\frac{x y^{-2}}{4x} = \frac{1}{4} \cdot \frac{x}{x} \cdot y^{-2} = \frac{1}{4} y^{-2}
\]

Now apply the exponent \(-2\):

\[
\left( \frac{1}{4} y^{-2} \right)^{-2} = \left( \frac{1}{4} \right)^{-2} \cdot (y^{-2})^{-2} = 4^2 \cdot y^4 = 16 y^4
\]

But don’t forget the negative sign in front:

\[
-16 y^4
\]

✔ Answer: \( \boxed{-16y^4} \)

---

Problem 7: \( \left( \frac{3}{x^3} \right)^2 \cdot x \)

First, square the fraction:

\[
\left( \frac{3}{x^3} \right)^2 = \frac{9}{x^6}
\]

Now multiply by \( x \):

\[
\frac{9}{x^6} \cdot x = \frac{9x}{x^6} = \frac{9}{x^5}
\]

✔ Answer: \( \boxed{\dfrac{9}{x^5}} \)

---

Problem 8: \( \left( \frac{1}{(3x)^{-3}} \right)^2 \)

First, simplify inside: \( \frac{1}{(3x)^{-3}} = (3x)^3 \)

Then square it:

\[
\left( (3x)^3 \right)^2 = (3x)^6 = 3^6 \cdot x^6 = 729 x^6
\]

✔ Answer: \( \boxed{729x^6} \)

---

Problem 9: \( \frac{(1 - y)^3}{y^3} \)

This expression has no negative exponents to simplify. The numerator is a binomial cubed, but unless asked to expand, we leave it as is.

✔ Answer: \( \boxed{\dfrac{(1 - y)^3}{y^3}} \)

*(Note: If you were to expand, it would be \( \frac{1 - 3y + 3y^2 - y^3}{y^3} \), but that’s not required here.)*

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Problem 10: \( \frac{1}{(t^3)^2} \cdot \frac{t^5}{2t} \)

Simplify each part:

First term: \( \frac{1}{(t^3)^2} = \frac{1}{t^6} \)

Second term: \( \frac{t^5}{2t} = \frac{1}{2} \cdot t^{5-1} = \frac{1}{2} t^4 \)

Now multiply:

\[
\frac{1}{t^6} \cdot \frac{1}{2} t^4 = \frac{1}{2} \cdot \frac{t^4}{t^6} = \frac{1}{2} t^{-2} = \frac{1}{2t^2}
\]

✔ Answer: \( \boxed{\dfrac{1}{2t^2}} \)

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## ✔ Final Answers Summary:

1. \( \boxed{-\dfrac{x^3}{216}} \)
2. \( \boxed{\dfrac{16y^{12}}{x^8}} \)
3. \( \boxed{\dfrac{1}{(18 + a)^2}} \)
4. \( \boxed{64(x+2)^2} \)
5. \( \boxed{\dfrac{s^{10}}{t^6}} \)
6. \( \boxed{-16y^4} \)
7. \( \boxed{\dfrac{9}{x^5}} \)
8. \( \boxed{729x^6} \)
9. \( \boxed{\dfrac{(1 - y)^3}{y^3}} \)
10. \( \boxed{\dfrac{1}{2t^2}} \)

Let me know if you’d like to see any expanded forms or step-by-step breakdowns for specific problems!