Simplifying Expressions With Negative Exponents Worksheet With ... - Free Printable

Let's solve each problem on the "Simplifying Expressions with Negative Exponents Worksheet 1" step by step, applying the negative exponent rule and other exponent rules.

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🔷 Negative Exponent Rule:

$$
a^{-n} = \frac{1}{a^n}, \quad \text{and} \quad \frac{1}{a^{-n}} = a^n
$$

Also recall:
- $ (ab)^n = a^n b^n $
- $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
- $ a^m \cdot a^n = a^{m+n} $
- $ \frac{a^m}{a^n} = a^{m-n} $

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

Apply the negative exponent rule:
$$
= \left( \frac{x}{-6} \right)^3 = \frac{x^3}{(-6)^3} = \frac{x^3}{-216} = -\frac{x^3}{216}
$$

✔ Answer: $ -\frac{x^3}{216} $

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

First simplify inside the parentheses:
$$
\frac{2x^{-2}}{y^{-3}} = 2 \cdot x^{-2} \cdot y^3 = 2y^3 x^{-2}
$$

Now raise to the 4th power:
$$
(2y^3 x^{-2})^4 = 2^4 \cdot (y^3)^4 \cdot (x^{-2})^4 = 16 \cdot y^{12} \cdot x^{-8}
$$

Write with positive exponents:
$$
= \frac{16y^{12}}{x^8}
$$

✔ Answer: $ \frac{16y^{12}}{x^8} $

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✔ Problem 3: $ (18 + a)^{-2} $

Use the negative exponent rule:
$$
= \frac{1}{(18 + a)^2}
$$

✔ Answer: $ \frac{1}{(18 + a)^2} $

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

Apply the negative exponent rule to the entire fraction:
$$
= \frac{(x+2)^2}{2^{-6}} = (x+2)^2 \cdot 2^6 = (x+2)^2 \cdot 64
$$

✔ Answer: $ 64(x+2)^2 $

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✔ Problem 5: $ (st^{-3} \cdot \frac{1}{s^{-4}})^2 $

First simplify inside the parentheses:

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

So:
$$
st^{-3} \cdot s^4 = s^{1+4} \cdot t^{-3} = s^5 t^{-3}
$$

Now square it:
$$
(s^5 t^{-3})^2 = s^{10} t^{-6} = \frac{s^{10}}{t^6}
$$

✔ Answer: $ \frac{s^{10}}{t^6} $

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

First simplify inside the parentheses:
$$
\frac{xy^{-2}}{4x} = \frac{y^{-2}}{4} = \frac{1}{4y^2}
$$

Now apply the exponent $-2$:
$$
-\left( \frac{1}{4y^2} \right)^{-2} = -\left( \frac{4y^2}{1} \right)^2 = -(4y^2)^2 = -16y^4
$$

✔ Answer: $ -16y^4 $

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✔ 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: $ \frac{9}{x^5} $

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

First simplify inside:
$$
\frac{1}{(3x)^{-3}} = (3x)^3 = 27x^3
$$

Now square:
$$
(27x^3)^2 = 729x^6
$$

✔ Answer: $ 729x^6 $

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

This expression is already simplified. But we can write it as:
$$
\left( \frac{1 - y}{y} \right)^3
$$

But unless asked, this form isn't required. The original is acceptable.

✔ Answer: $ \frac{(1 - y)^3}{y^3} $

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

First simplify each part:

- $ (t^3)^2 = t^6 $, so $ \frac{1}{t^6} $
- $ \frac{t^5}{2t} = \frac{t^4}{2} $

Now multiply:
$$
\frac{1}{t^6} \cdot \frac{t^4}{2} = \frac{t^4}{2t^6} = \frac{1}{2t^2}
$$

✔ Answer: $ \frac{1}{2t^2} $

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✔ Final Answers Summary:

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

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