Simplifying Expressions worksheet featuring ten algebraic problems for practice.

Let's go through each expression one by one and simplify them step by step.

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1. $ u \cdot \left( \frac{81u^2}{9u} \right) $

First, simplify the fraction inside:
$$
\frac{81u^2}{9u} = \frac{81}{9} \cdot \frac{u^2}{u} = 9 \cdot u^{2-1} = 9u
$$

Now multiply by $ u $:
$$
u \cdot 9u = 9u^2
$$

✔ Answer: $ \boxed{9u^2} $

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2. $ v^2 \cdot \left( -\frac{4}{4} \right) $

Simplify the fraction:
$$
-\frac{4}{4} = -1
$$

So:
$$
v^2 \cdot (-1) = -v^2
$$

✔ Answer: $ \boxed{-v^2} $

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3. $ \frac{-126z^2}{7z^2 \cdot 9} $

First, simplify denominator:
$$
7z^2 \cdot 9 = 63z^2
$$

Now:
$$
\frac{-126z^2}{63z^2} = \frac{-126}{63} \cdot \frac{z^2}{z^2} = -2 \cdot 1 = -2
$$

✔ Answer: $ \boxed{-2} $

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4. $ \frac{-3b^2}{-b} \cdot (-9b^2) $

First, simplify $ \frac{-3b^2}{-b} $:
$$
\frac{-3b^2}{-b} = \frac{3b^2}{b} = 3b
$$

Now multiply by $ -9b^2 $:
$$
3b \cdot (-9b^2) = -27b^{1+2} = -27b^3
$$

✔ Answer: $ \boxed{-27b^3} $

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5. $ \frac{16x^2}{-8x^2} \cdot x $

Simplify the fraction:
$$
\frac{16x^2}{-8x^2} = \frac{16}{-8} \cdot \frac{x^2}{x^2} = -2 \cdot 1 = -2
$$

Now multiply by $ x $:
$$
-2 \cdot x = -2x
$$

✔ Answer: $ \boxed{-2x} $

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6. $ b^2 \cdot \frac{2b^3}{2b} $

Simplify the fraction:
$$
\frac{2b^3}{2b} = \frac{2}{2} \cdot \frac{b^3}{b} = 1 \cdot b^{3-1} = b^2
$$

Now multiply by $ b^2 $:
$$
b^2 \cdot b^2 = b^{2+2} = b^4
$$

✔ Answer: $ \boxed{b^4} $

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7. $ 5x - 6x - x $

Combine like terms:
$$
(5 - 6 - 1)x = (-2)x = -2x
$$

✔ Answer: $ \boxed{-2x} $

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8. $ -8 \cdot 5y \cdot y^2 $

Multiply constants and variables:
$$
-8 \cdot 5 = -40
$$
$$
y \cdot y^2 = y^{1+2} = y^3
$$

So:
$$
-40y^3
$$

✔ Answer: $ \boxed{-40y^3} $

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9. $ \frac{81u^6}{-9u^2 \cdot 3u^2} $

First, simplify denominator:
$$
-9u^2 \cdot 3u^2 = (-9 \cdot 3)(u^2 \cdot u^2) = -27u^4
$$

Now:
$$
\frac{81u^6}{-27u^4} = \frac{81}{-27} \cdot \frac{u^6}{u^4} = -3 \cdot u^{6-4} = -3u^2
$$

✔ Answer: $ \boxed{-3u^2} $

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10. $ -y \cdot \left( -\frac{2y}{-y} \right) $

First, simplify inside the parentheses:
$$
-\frac{2y}{-y} = \frac{-2y}{-y} = \frac{2y}{y} = 2
$$

Now multiply:
$$
-y \cdot 2 = -2y
$$

✔ Answer: $ \boxed{-2y} $

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✔ Final Answers:

1. $ 9u^2 $
2. $ -v^2 $
3. $ -2 $
4. $ -27b^3 $
5. $ -2x $
6. $ b^4 $
7. $ -2x $
8. $ -40y^3 $
9. $ -3u^2 $
10. $ -2y $

Let me know if you'd like these explained in a different format or with more steps!