Math worksheet for simplifying expressions with negative exponents and negative bases.

Sure! Let’s solve each problem step by step, using the rules of negative exponents:

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

Also, remember:
- When simplifying fractions with negative exponents, move terms with negative exponents to the opposite part of the fraction (numerator ↔ denominator) and make the exponent positive.
- For powers of powers: \( (a^m)^n = a^{m \cdot n} \)
- For fractions raised to a power: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \)

---

Problem 1: \( -\frac{15}{x^{-3}} \)

Move \( x^{-3} \) to numerator → becomes \( x^3 \):

\[
-\frac{15}{x^{-3}} = -15 \cdot x^3 = \boxed{-15x^3}
\]

---

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

Move \( x^{-2} \) to numerator → becomes \( x^2 \):

\[
\frac{2x^4}{-3x^{-2}} = \frac{2x^4 \cdot x^2}{-3} = \frac{2x^{6}}{-3} = \boxed{-\frac{2}{3}x^6}
\]

---

Problem 3: \( -2^{-5} \)

Note: Only the 2 is raised to the power. The negative sign is outside.

\[
-2^{-5} = -\left( \frac{1}{2^5} \right) = -\frac{1}{32} = \boxed{-\frac{1}{32}}
\]

---

Problem 4: \( -\frac{6x^{-4}}{3x} \)

Simplify coefficients: \( \frac{6}{3} = 2 \), so:

\[
-\frac{6x^{-4}}{3x} = -2 \cdot \frac{x^{-4}}{x} = -2 \cdot x^{-4 - 1} = -2x^{-5}
\]

Now write with positive exponent:

\[
-2x^{-5} = \boxed{-\frac{2}{x^5}}
\]

---

Problem 5: \( \frac{3y^{-4}}{y^{-2}} \)

Subtract exponents: \( y^{-4 - (-2)} = y^{-2} \)

\[
\frac{3y^{-4}}{y^{-2}} = 3y^{-2} = \boxed{\frac{3}{y^2}}
\]

---

Problem 6: \( -\frac{8}{a^{-2}b^{-3}} \)

Move both \( a^{-2} \) and \( b^{-3} \) to numerator → become \( a^2 \) and \( b^3 \):

\[
-\frac{8}{a^{-2}b^{-3}} = -8 \cdot a^2 \cdot b^3 = \boxed{-8a^2b^3}
\]

---

Problem 7: \( \left( \frac{x^{-6}}{x^{-2}} \right)^{-2} \)

First simplify inside parentheses:

\[
\frac{x^{-6}}{x^{-2}} = x^{-6 - (-2)} = x^{-4}
\]

Now raise to power -2:

\[
(x^{-4})^{-2} = x^{(-4)(-2)} = x^8 = \boxed{x^8}
\]

---

Problem 8: \( -\frac{12x}{4x^{-3}} \)

Simplify coefficients: \( \frac{12}{4} = 3 \)

\[
-\frac{12x}{4x^{-3}} = -3 \cdot \frac{x}{x^{-3}} = -3 \cdot x^{1 - (-3)} = -3x^4 = \boxed{-3x^4}
\]

---

Problem 9: \( \frac{7b}{-9b^{-2}} \)

Move \( b^{-2} \) to numerator → becomes \( b^2 \):

\[
\frac{7b}{-9b^{-2}} = \frac{7b \cdot b^2}{-9} = \frac{7b^3}{-9} = \boxed{-\frac{7}{9}b^3}
\]

---

Problem 10: \( \frac{5p^{-3}}{25n^{-3}} \)

Simplify coefficients: \( \frac{5}{25} = \frac{1}{5} \)

Move \( p^{-3} \) to denominator → \( p^3 \); move \( n^{-3} \) to numerator → \( n^3 \)

\[
\frac{5p^{-3}}{25n^{-3}} = \frac{1}{5} \cdot \frac{n^3}{p^3} = \boxed{\frac{n^3}{5p^3}}
\]

---

Problem 11: \( \frac{-25}{x^{-6}} \)

Move \( x^{-6} \) to numerator → becomes \( x^6 \)

\[
\frac{-25}{x^{-6}} = -25x^6 = \boxed{-25x^6}
\]

---

Problem 12: \( \left( \frac{8x^{-2}}{2x^2} \right)^{-3} \)

First simplify inside:

\[
\frac{8x^{-2}}{2x^2} = 4x^{-2 - 2} = 4x^{-4}
\]

Now raise to -3:

\[
(4x^{-4})^{-3} = 4^{-3} \cdot x^{12} = \frac{1}{64}x^{12} = \boxed{\frac{x^{12}}{64}}
\]

---

Problem 13: \( w^{-5} \)

Apply rule directly:

\[
w^{-5} = \boxed{\frac{1}{w^5}}
\]

---

Problem 14: \( 5c^{-3} \)

\[
5c^{-3} = \boxed{\frac{5}{c^3}}
\]

---

Problem 15: \( \frac{1}{-x^{-2}} \)

Move \( x^{-2} \) to numerator → becomes \( x^2 \), keep negative sign:

\[
\frac{1}{-x^{-2}} = -x^2 = \boxed{-x^2}
\]

---

Problem 16: \( \left( \frac{-2}{-5x^{-2}} \right)^{-1} \)

First simplify inside:

\[
\frac{-2}{-5x^{-2}} = \frac{2}{5x^{-2}} = \frac{2}{5} \cdot x^2
\]

Now raise to -1 → reciprocal:

\[
\left( \frac{2}{5}x^2 \right)^{-1} = \frac{5}{2} \cdot x^{-2} = \boxed{\frac{5}{2x^2}}
\]

---

Problem 17: \( \left( \frac{5x}{yx^2} \right)^{-2} \)

Simplify inside first:

\[
\frac{5x}{yx^2} = \frac{5}{y} \cdot x^{1 - 2} = \frac{5}{y} \cdot x^{-1}
\]

Now raise to -2:

\[
\left( \frac{5}{y} \cdot x^{-1} \right)^{-2} = \left( \frac{5}{y} \right)^{-2} \cdot (x^{-1})^{-2} = \frac{y^2}{25} \cdot x^2 = \boxed{\frac{x^2 y^2}{25}}
\]

---

Problem 18: \( \frac{-3}{x^{-2}} \)

Move \( x^{-2} \) to numerator → becomes \( x^2 \)

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

---

Problem 19: \( b^{-2} \)

\[
b^{-2} = \boxed{\frac{1}{b^2}}
\]

---

Problem 20: \( \left( \frac{-3}{5} \right)^{-2} \)

Reciprocal and square:

\[
\left( \frac{-3}{5} \right)^{-2} = \left( \frac{5}{-3} \right)^2 = \frac{25}{9} = \boxed{\frac{25}{9}}
\]

*(Note: Even power makes negative positive)*

---

Problem 21: \( -z^{-4} \)

Only z is raised to power; negative sign outside.

\[
-z^{-4} = -\frac{1}{z^4} = \boxed{-\frac{1}{z^4}}
\]

---

Problem 22: \( 5x^{-4}y^{-2} \)

Write with positive exponents:

\[
5x^{-4}y^{-2} = \boxed{\frac{5}{x^4 y^2}}
\]

---

## ✔ Final Answers Summary:

1) \( \boxed{-15x^3} \)
2) \( \boxed{-\frac{2}{3}x^6} \)
3) \( \boxed{-\frac{1}{32}} \)
4) \( \boxed{-\frac{2}{x^5}} \)
5) \( \boxed{\frac{3}{y^2}} \)
6) \( \boxed{-8a^2b^3} \)
7) \( \boxed{x^8} \)
8) \( \boxed{-3x^4} \)
9) \( \boxed{-\frac{7}{9}b^3} \)
10) \( \boxed{\frac{n^3}{5p^3}} \)
11) \( \boxed{-25x^6} \)
12) \( \boxed{\frac{x^{12}}{64}} \)
13) \( \boxed{\frac{1}{w^5}} \)
14) \( \boxed{\frac{5}{c^3}} \)
15) \( \boxed{-x^2} \)
16) \( \boxed{\frac{5}{2x^2}} \)
17) \( \boxed{\frac{x^2 y^2}{25}} \)
18) \( \boxed{-3x^2} \)
19) \( \boxed{\frac{1}{b^2}} \)
20) \( \boxed{\frac{25}{9}} \)
21) \( \boxed{-\frac{1}{z^4}} \)
22) \( \boxed{\frac{5}{x^4 y^2}} \)

Let me know if you’d like a printable version or explanation for any specific problem!