Math worksheet focused on the distributive property, including expressions to classify and simplify.

Let's solve the Distributive Property Worksheet step by step.

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Part 1: For each expression below, indicate whether or not you could simplify it using the distributive property.

We are to determine if the distributive property applies — that is, whether we can distribute a factor across terms inside parentheses.

The distributive property states:
> $ a(b + c) = ab + ac $

It applies when there’s a number or variable multiplied by a sum or difference in parentheses.

Let’s go through each:

#### a) $ 3(4 + y) $
- Yes → This is a number (3) times a sum $(4 + y)$
- ✔ Yes, distributive property applies

#### b) $ 7(x + 2) $
- Yes → 7 times a sum
- ✔ Yes

#### c) $ (z + 6) - 7 $
- No → There's no multiplication over a sum; this is subtraction of 7 from a sum.
- ✘ No

#### d) $ (c - 2) + 6 $
- No → Again, just addition and subtraction; no multiplication across parentheses.
- ✘ No

#### e) $ c - 2 + 6 $
- No → This is just simplifying like terms (no parentheses, no distribution)
- ✘ No

#### f) $ 5(5 - 5h) $
- Yes → 5 times a difference
- ✔ Yes

#### g) $ 9(a + k + g) $
- Yes → 9 times a sum of three terms
- ✔ Yes

#### h) $ 9 + (a + k + g) $
- No → Addition outside the parentheses, no multiplication
- ✘ No

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✔ Final Answers for Part 1:

| Expression | Distrib. Prop.? |
|------------------------|------------------|
| 3(4 + y) | Yes |
| 7(x + 2) | Yes |
| (z + 6) - 7 | No |
| (c - 2) + 6 | No |
| c - 2 + 6 | No |
| 5(5 - 5h) | Yes |
| 9(a + k + g) | Yes |
| 9 + (a + k + g) | No |

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Part 2: Simplify each expression below using the distributive property.

We apply the distributive property: $ a(b + c) = ab + ac $

#### a) $ 9(3 + 4) $
- $ = 9 \cdot 3 + 9 \cdot 4 = 27 + 36 = \boxed{63} $

#### b) $ (6 - 2) \cdot 3 $
- First simplify inside: $ 6 - 2 = 4 $, then $ 4 \cdot 3 = 12 $
- Or use distributive: $ 6\cdot3 - 2\cdot3 = 18 - 6 = \boxed{12} $

#### c) $ 7 \cdot (5 + 5 - 8) $
- Simplify inside first: $ 5 + 5 - 8 = 2 $, so $ 7 \cdot 2 = \boxed{14} $
- Alternatively: $ 7\cdot5 + 7\cdot5 - 7\cdot8 = 35 + 35 - 56 = 70 - 56 = \boxed{14} $

#### d) $ (7 + y) \cdot 2 $
- Distribute 2: $ 7\cdot2 + y\cdot2 = 14 + 2y $
- Answer: $ \boxed{14 + 2y} $

#### e) $ 4(5 + 3g) $
- $ 4\cdot5 + 4\cdot3g = 20 + 12g $
- Answer: $ \boxed{20 + 12g} $

#### f) $ 6(4 - w) $
- $ 6\cdot4 - 6\cdot w = 24 - 6w $
- Answer: $ \boxed{24 - 6w} $

#### g) $ (3f - 6) \cdot 9 $
- Distribute 9: $ 9\cdot3f - 9\cdot6 = 27f - 54 $
- Answer: $ \boxed{27f - 54} $

#### h) $ (9 + 8a - 4a) + 4 $
- Wait! This one is tricky. The distributive property does not apply directly here because there's no multiplication over parentheses.
- But let's simplify:
- Combine like terms: $ 9 + (8a - 4a) + 4 = 9 + 4a + 4 = 13 + 4a $
- So: $ \boxed{13 + 4a} $
- Note: This doesn't require distributive property; just combining like terms.

Wait — the instruction says "simplify using the distributive property". But here, there's no multiplication over a group, so distributive property does not apply. Let's check again.

Actually, the expression is:
$ (9 + 8a - 4a) + 4 $

This is already simplified within the parentheses: $ 9 + 4a $, then add 4 → $ 13 + 4a $

But since no distribution was needed, perhaps it's a mistake? Let's recheck the original.

Wait — maybe it's meant to be something like $ 9(1 + 8a - 4a) $? But as written, it's just $ (9 + 8a - 4a) + 4 $

So unless there's a typo, this expression cannot be simplified using the distributive property — but we can still simplify it algebraically.

So answer: $ \boxed{13 + 4a} $ (but not via distributive property)

Alternatively, maybe the problem expects us to recognize that it's not applicable?

But let's assume the expression is correct.

Wait — look at part i:

#### i) $ 7 \cdot (3 - 12a) $
- Apply distributive property: $ 7\cdot3 - 7\cdot12a = 21 - 84a $
- Answer: $ \boxed{21 - 84a} $

So back to h): $ (9 + 8a - 4a) + 4 $

We can write it as:
$ (9 + 4a) + 4 = 13 + 4a $

But again, no distribution involved.

So likely, the distributive property does not apply here. But since the question asks to simplify using the distributive property, perhaps it's a trick?

Alternatively, maybe the expression is supposed to be: $ 9(1 + 8a - 4a) $? That would make sense.

But based on what's written: $ (9 + 8a - 4a) + 4 $, we just simplify:

- $ 8a - 4a = 4a $
- $ 9 + 4a + 4 = 13 + 4a $

So final answer: $ \boxed{13 + 4a} $

Even though distributive property isn't used, it's the simplified form.

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Part 3: How about fractions and decimals?

#### j) $ 4.6(3x + 2) $
- Distribute 4.6:
- $ 4.6 \cdot 3x = 13.8x $
- $ 4.6 \cdot 2 = 9.2 $
- Answer: $ \boxed{13.8x + 9.2} $

#### k) $ \frac{1}{2}(10 - 14t) $
- Distribute $ \frac{1}{2} $:
- $ \frac{1}{2} \cdot 10 = 5 $
- $ \frac{1}{2} \cdot (-14t) = -7t $
- Answer: $ \boxed{5 - 7t} $

#### l) $ \frac{1}{2}\left(\frac{1}{2}m + \frac{1}{4}\right) $
- Distribute $ \frac{1}{2} $:
- $ \frac{1}{2} \cdot \frac{1}{2}m = \frac{1}{4}m $
- $ \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} $
- Answer: $ \boxed{\frac{1}{4}m + \frac{1}{8}} $

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✔ Final Answers Summary:

#### Part 1:
| Expression | Distrib. Prop.? |
|------------------------|------------------|
| 3(4 + y) | Yes |
| 7(x + 2) | Yes |
| (z + 6) - 7 | No |
| (c - 2) + 6 | No |
| c - 2 + 6 | No |
| 5(5 - 5h) | Yes |
| 9(a + k + g) | Yes |
| 9 + (a + k + g) | No |

#### Part 2:
a) $ 9(3 + 4) = \boxed{63} $
b) $ (6 - 2) \cdot 3 = \boxed{12} $
c) $ 7 \cdot (5 + 5 - 8) = \boxed{14} $
d) $ (7 + y) \cdot 2 = \boxed{14 + 2y} $
e) $ 4(5 + 3g) = \boxed{20 + 12g} $
f) $ 6(4 - w) = \boxed{24 - 6w} $
g) $ (3f - 6) \cdot 9 = \boxed{27f - 54} $
h) $ (9 + 8a - 4a) + 4 = \boxed{13 + 4a} $
i) $ 7 \cdot (3 - 12a) = \boxed{21 - 84a} $

#### Part 3:
j) $ 4.6(3x + 2) = \boxed{13.8x + 9.2} $
k) $ \frac{1}{2}(10 - 14t) = \boxed{5 - 7t} $
l) $ \frac{1}{2}\left(\frac{1}{2}m + \frac{1}{4}\right) = \boxed{\frac{1}{4}m + \frac{1}{8}} $

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