Simplifying expressions with negative exponents worksheet for algebra practice.

Let's solve each problem step by step, simplifying expressions with negative exponents and ensuring the final answer contains only positive exponents.

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Problem 1:

$$
2m^{-1}n^{-3} \cdot (2m^{-1}n^{-3})^4
$$

Step 1: Simplify the exponent part:
$$
(2m^{-1}n^{-3})^4 = 2^4 \cdot m^{-4} \cdot n^{-12} = 16m^{-4}n^{-12}
$$

Step 2: Multiply by the first term:
$$
2m^{-1}n^{-3} \cdot 16m^{-4}n^{-12} = (2 \cdot 16) \cdot m^{-1 + (-4)} \cdot n^{-3 + (-12)} = 32m^{-5}n^{-15}
$$

Step 3: Convert negative exponents to positive:
$$
= \frac{32}{m^5n^{15}}
$$

✔ Answer: $\boxed{\dfrac{32}{m^5n^{15}}}$

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Problem 2:

$$
\frac{2x^{-3}}{(x^4y^{-3})^{-1}}
$$

Step 1: Simplify the denominator:
$$
(x^4y^{-3})^{-1} = x^{-4}y^{3}
$$

So the expression becomes:
$$
\frac{2x^{-3}}{x^{-4}y^3}
$$

Step 2: Divide powers:
$$
= 2 \cdot x^{-3 - (-4)} \cdot y^{-3} = 2x^{1}y^{-3} = 2x \cdot \frac{1}{y^3}
$$

✔ Answer: $\boxed{\dfrac{2x}{y^3}}$

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Problem 3:

$$
\frac{4x^{-5}y^3 \cdot 3x^3y^{-2}}{6x^{-5}y^3}
$$

Step 1: Combine numerator:
$$
(4 \cdot 3) \cdot x^{-5+3} \cdot y^{3 + (-2)} = 12x^{-2}y^1
$$

Now divide by denominator:
$$
\frac{12x^{-2}y}{6x^{-5}y^3} = \frac{12}{6} \cdot x^{-2 - (-5)} \cdot y^{1 - 3} = 2x^{3}y^{-2}
$$

Convert negative exponent:
$$
= 2x^3 \cdot \frac{1}{y^2}
$$

✔ Answer: $\boxed{\dfrac{2x^3}{y^2}}$

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Problem 4:

$$
\frac{a^8b^{-5}}{a^2b^3}
$$

Step 1: Subtract exponents:
$$
a^{8-2} \cdot b^{-5 - 3} = a^6b^{-8}
$$

Convert negative exponent:
$$
= \frac{a^6}{b^8}
$$

✔ Answer: $\boxed{\dfrac{a^6}{b^8}}$

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Problem 5:

$$
\left(\frac{4x}{12x^2y}\right)^{-2}
$$

Step 1: Simplify inside the parentheses:
$$
\frac{4x}{12x^2y} = \frac{1}{3xy}
$$

Now apply the exponent:
$$
\left(\frac{1}{3xy}\right)^{-2} = \frac{1}{(3xy)^{-2}} = (3xy)^2 = 9x^2y^2
$$

✔ Answer: $\boxed{9x^2y^2}$

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Problem 6:

$$
\left(\frac{y}{x^2}\right)^{-5}
$$

Apply negative exponent rule:
$$
= \left(\frac{x^2}{y}\right)^5 = \frac{x^{10}}{y^5}
$$

✔ Answer: $\boxed{\dfrac{x^{10}}{y^5}}$

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Problem 7:

$$
\frac{w(xy)^{-2}}{(3tv)^2}
$$

Step 1: Expand:
$$
(xy)^{-2} = x^{-2}y^{-2}, \quad (3tv)^2 = 9t^2v^2
$$

So:
$$
\frac{w \cdot x^{-2}y^{-2}}{9t^2v^2} = \frac{w}{9t^2v^2x^2y^2}
$$

✔ Answer: $\boxed{\dfrac{w}{9t^2v^2x^2y^2}}$

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Problem 8:

$$
\frac{(xy)^{-1}}{ab^{-2}}
$$

Step 1: Rewrite $(xy)^{-1} = \frac{1}{xy}$

So:
$$
\frac{1}{xy \cdot ab^{-2}} = \frac{1}{xyab^{-2}}
$$

Now simplify $b^{-2}$ in denominator → move to numerator as $b^2$:
$$
= \frac{b^2}{xya}
$$

✔ Answer: $\boxed{\dfrac{b^2}{axy}}$

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✔ Final Answers:

1. $\boxed{\dfrac{32}{m^5n^{15}}}$
2. $\boxed{\dfrac{2x}{y^3}}$
3. $\boxed{\dfrac{2x^3}{y^2}}$
4. $\boxed{\dfrac{a^6}{b^8}}$
5. $\boxed{9x^2y^2}$
6. $\boxed{\dfrac{x^{10}}{y^5}}$
7. $\boxed{\dfrac{w}{9t^2v^2x^2y^2}}$
8. $\boxed{\dfrac{b^2}{axy}}$

Let me know if you'd like these explained further!