Algebra worksheet focusing on simplifying expressions with negative and zero exponents.

Let's solve each problem step by step, applying the rules of exponents, especially those involving negative and zero exponents.

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🔷 Rules to Remember:

1. $ a^{-n} = \frac{1}{a^n} $
2. $ \frac{1}{a^{-n}} = a^n $
3. $ a^0 = 1 $ for any $ a \neq 0 $
4. $ (ab)^n = a^n b^n $
5. $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
6. $ a^m \cdot a^n = a^{m+n} $
7. $ \frac{a^m}{a^n} = a^{m-n} $

We'll simplify each expression so that only positive exponents remain.

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✔ Solutions:

#### 1) $ 2^{-2} $
$$
= \frac{1}{2^2} = \frac{1}{4}
$$

#### 2) $ 8y^{-4} $
$$
= \frac{8}{y^4}
$$

#### 3) $ 3yx^{-2} $
$$
= \frac{3y}{x^2}
$$

#### 4) $ x^2 y^{-2} $
$$
= \frac{x^2}{y^2}
$$

#### 5) $ \frac{9a^{-5}}{3a^5} $
Simplify coefficients and apply exponent rules:
$$
= \frac{9}{3} \cdot \frac{a^{-5}}{a^5} = 3 \cdot a^{-5-5} = 3a^{-10} = \frac{3}{a^{10}}
$$

#### 6) $ \frac{1}{5^{-2}} $
$$
= 5^2 = 25
$$

#### 7) $ 2yx^{-2} $
$$
= \frac{2y}{x^2}
$$

#### 8) $ 10^{-3} \cdot b^5 $
$$
= \frac{1}{10^3} \cdot b^5 = \frac{b^5}{1000}
$$

#### 9) $ 7m^{-2} $
$$
= \frac{7}{m^2}
$$

#### 10) $ 3b^4 \cdot b^2 $
$$
= 3b^{4+2} = 3b^6
$$

#### 11) $ 3x^{-5} \cdot 4x $
$$
= (3 \cdot 4) \cdot x^{-5+1} = 12x^{-4} = \frac{12}{x^4}
$$

#### 12) $ \frac{4x^{-4}y^{-5}}{2y} $
$$
= \frac{4}{2} \cdot x^{-4} \cdot y^{-5-1} = 2x^{-4}y^{-6} = \frac{2}{x^4 y^6}
$$

#### 13) $ \frac{v^3}{v^5} $
$$
= v^{3-5} = v^{-2} = \frac{1}{v^2}
$$

#### 14) $ (4x^{-2}y^5)^{-3} $
Apply power to each part:
$$
= 4^{-3} \cdot x^{(-2)(-3)} \cdot y^{5 \cdot (-3)} = \frac{1}{64} \cdot x^6 \cdot y^{-15} = \frac{x^6}{64y^{15}}
$$

#### 15) $ 3^{-4} $
$$
= \frac{1}{3^4} = \frac{1}{81}
$$

#### 16) $ 10v^{-5} \cdot 8v^3 $
$$
= (10 \cdot 8) \cdot v^{-5+3} = 80v^{-2} = \frac{80}{v^2}
$$

#### 17) $ (x^{-4}y^4)^3 $
$$
= x^{-12} y^{12} = \frac{y^{12}}{x^{12}}
$$

#### 18) $ (3uv)^{-2} $
$$
= 3^{-2} u^{-2} v^{-2} = \frac{1}{9u^2 v^2}
$$

#### 19) $ \frac{3u^{-4}v^2}{3u^{-3}v^2} $
Cancel common terms:
$$
= \frac{3}{3} \cdot \frac{u^{-4}}{u^{-3}} \cdot \frac{v^2}{v^2} = 1 \cdot u^{-4 - (-3)} \cdot 1 = u^{-1} = \frac{1}{u}
$$

#### 20) $ 5m^{-2}n^{-4} $
$$
= \frac{5}{m^2 n^4}
$$

#### 21) $ 5nm^{-4} $
$$
= \frac{5n}{m^4}
$$

#### 22) $ 5n^3 \cdot 8n^3 $
$$
= (5 \cdot 8) \cdot n^{3+3} = 40n^6
$$

#### 23) $ (u^2 v^{-4})^4 $
$$
= u^{8} v^{-16} = \frac{u^8}{v^{16}}
$$

#### 24) $ \frac{6u^3 v^{-3}}{3u^4 v^3} $
$$
= \frac{6}{3} \cdot u^{3-4} \cdot v^{-3-3} = 2 \cdot u^{-1} \cdot v^{-6} = \frac{2}{u v^6}
$$

#### 25) $ (5a^4)^0 $
Any nonzero expression to the 0 power is 1:
$$
= 1
$$

#### 26) $ 3r^{-4} $
$$
= \frac{3}{r^4}
$$

#### 27) $ 4x^{-4} $
$$
= \frac{4}{x^4}
$$

#### 28) $ \frac{6x^2 y^{-1}}{10x^0} $
Note: $ x^0 = 1 $
$$
= \frac{6x^2}{10 y} = \frac{3x^2}{5y}
$$

#### 29) $ \frac{4x^{-3} y^4}{7x^2 y^{-4}} $
$$
= \frac{4}{7} \cdot x^{-3-2} \cdot y^{4 - (-4)} = \frac{4}{7} x^{-5} y^8 = \frac{4y^8}{7x^5}
$$

#### 30) $ \frac{3yx^4}{6x^3} $
$$
= \frac{3}{6} \cdot y \cdot x^{4-3} = \frac{1}{2} y x = \frac{xy}{2}
$$

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🔢 Evaluate Expressions:

#### 31) Find the value of $ 7x^0 - (6x)^0 $

Recall: $ a^0 = 1 $ for $ a \neq 0 $

So:
- $ x^0 = 1 $
- $ (6x)^0 = 1 $ (as long as $ x \neq 0 $)

Then:
$$
7(1) - 1 = 7 - 1 = \boxed{6}
$$

#### 32) Find the value of $ 5x^0 + 2x^{-1} $ when $ x = 4 $

First, $ x^0 = 1 $, $ x^{-1} = \frac{1}{x} = \frac{1}{4} $

So:
$$
5(1) + 2\left(\frac{1}{4}\right) = 5 + \frac{2}{4} = 5 + \frac{1}{2} = \boxed{5.5} \text{ or } \boxed{\frac{11}{2}}
$$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{1}{4} $ |
| 2 | $ \frac{8}{y^4} $ |
| 3 | $ \frac{3y}{x^2} $ |
| 4 | $ \frac{x^2}{y^2} $ |
| 5 | $ \frac{3}{a^{10}} $ |
| 6 | $ 25 $ |
| 7 | $ \frac{2y}{x^2} $ |
| 8 | $ \frac{b^5}{1000} $ |
| 9 | $ \frac{7}{m^2} $ |
| 10 | $ 3b^6 $ |
| 11 | $ \frac{12}{x^4} $ |
| 12 | $ \frac{2}{x^4 y^6} $ |
| 13 | $ \frac{1}{v^2} $ |
| 14 | $ \frac{x^6}{64y^{15}} $ |
| 15 | $ \frac{1}{81} $ |
| 16 | $ \frac{80}{v^2} $ |
| 17 | $ \frac{y^{12}}{x^{12}} $ |
| 18 | $ \frac{1}{9u^2 v^2} $ |
| 19 | $ \frac{1}{u} $ |
| 20 | $ \frac{5}{m^2 n^4} $ |
| 21 | $ \frac{5n}{m^4} $ |
| 22 | $ 40n^6 $ |
| 23 | $ \frac{u^8}{v^{16}} $ |
| 24 | $ \frac{2}{u v^6} $ |
| 25 | $ 1 $ |
| 26 | $ \frac{3}{r^4} $ |
| 27 | $ \frac{4}{x^4} $ |
| 28 | $ \frac{3x^2}{5y} $ |
| 29 | $ \frac{4y^8}{7x^5} $ |
| 30 | $ \frac{xy}{2} $ |
| 31 | $ 6 $ |
| 32 | $ \frac{11}{2} $ or $ 5.5 $ |

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