SOLUTION: Simplifying algebraic expressions worksheet - Studypool - Free Printable

To simplify the given expressions, we will use the distributive property and combine like terms where applicable. Let's go through each expression step by step.

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1. Simplify \( 2t(3t + 2) + 5t \)

#### Step 1: Distribute \( 2t \) across \( (3t + 2) \)
\[
2t(3t + 2) = 2t \cdot 3t + 2t \cdot 2 = 6t^2 + 4t
\]

#### Step 2: Add \( 5t \)
\[
6t^2 + 4t + 5t = 6t^2 + 9t
\]

#### Final Answer:
\[
\boxed{6t^2 + 9t}
\]

---

2. Simplify \( 10k + 2k(-8k - 5) \)

#### Step 1: Distribute \( 2k \) across \( (-8k - 5) \)
\[
2k(-8k - 5) = 2k \cdot (-8k) + 2k \cdot (-5) = -16k^2 - 10k
\]

#### Step 2: Combine with \( 10k \)
\[
10k + (-16k^2 - 10k) = 10k - 16k^2 - 10k
\]

#### Step 3: Combine like terms
\[
10k - 10k - 16k^2 = -16k^2
\]

#### Final Answer:
\[
\boxed{-16k^2}
\]

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3. Simplify \( -m(2m + 3) - 8 \)

#### Step 1: Distribute \( -m \) across \( (2m + 3) \)
\[
-m(2m + 3) = -m \cdot 2m + (-m) \cdot 3 = -2m^2 - 3m
\]

#### Step 2: Subtract 8
\[
-2m^2 - 3m - 8
\]

#### Final Answer:
\[
\boxed{-2m^2 - 3m - 8}
\]

---

4. Simplify \( 3x - 3x(-7x + 9y - 2) \)

#### Step 1: Distribute \( -3x \) across \( (-7x + 9y - 2) \)
\[
-3x(-7x + 9y - 2) = -3x \cdot (-7x) + (-3x) \cdot 9y + (-3x) \cdot (-2)
\]
\[
= 21x^2 - 27xy + 6x
\]

#### Step 2: Combine with \( 3x \)
\[
3x + (21x^2 - 27xy + 6x) = 3x + 21x^2 - 27xy + 6x
\]

#### Step 3: Combine like terms
\[
3x + 6x = 9x
\]
\[
21x^2 - 27xy + 9x
\]

#### Final Answer:
\[
\boxed{21x^2 - 27xy + 9x}
\]

---

5. Simplify \( w(-3w + 5) - 5t \)

#### Step 1: Distribute \( w \) across \( (-3w + 5) \)
\[
w(-3w + 5) = w \cdot (-3w) + w \cdot 5 = -3w^2 + 5w
\]

#### Step 2: Subtract \( 5t \)
\[
-3w^2 + 5w - 5t
\]

#### Final Answer:
\[
\boxed{-3w^2 + 5w - 5t}
\]

---

6. Simplify \( -5w(-5 - 4y - 2w) - 10w^2 \)

#### Step 1: Distribute \( -5w \) across \( (-5 - 4y - 2w) \)
\[
-5w(-5 - 4y - 2w) = -5w \cdot (-5) + (-5w) \cdot (-4y) + (-5w) \cdot (-2w)
\]
\[
= 25w + 20wy + 10w^2
\]

#### Step 2: Subtract \( 10w^2 \)
\[
25w + 20wy + 10w^2 - 10w^2
\]

#### Step 3: Combine like terms
\[
10w^2 - 10w^2 = 0
\]
\[
25w + 20wy
\]

#### Final Answer:
\[
\boxed{25w + 20wy}
\]

---

7. Simplify \( -2z(-4p + 6z) + 8p \)

#### Step 1: Distribute \( -2z \) across \( (-4p + 6z) \)
\[
-2z(-4p + 6z) = -2z \cdot (-4p) + (-2z) \cdot 6z
\]
\[
= 8pz - 12z^2
\]

#### Step 2: Add \( 8p \)
\[
8pz - 12z^2 + 8p
\]

#### Final Answer:
\[
\boxed{8pz - 12z^2 + 8p}
\]

---

8. Simplify \( -4(7n - 9p - 3) + 4p \)

#### Step 1: Distribute \( -4 \) across \( (7n - 9p - 3) \)
\[
-4(7n - 9p - 3) = -4 \cdot 7n + (-4) \cdot (-9p) + (-4) \cdot (-3)
\]
\[
= -28n + 36p + 12
\]

#### Step 2: Add \( 4p \)
\[
-28n + 36p + 12 + 4p
\]

#### Step 3: Combine like terms
\[
36p + 4p = 40p
\]
\[
-28n + 40p + 12
\]

#### Final Answer:
\[
\boxed{-28n + 40p + 12}
\]

---

9. Simplify \( 10p + 5(6 + 7p) \)

#### Step 1: Distribute \( 5 \) across \( (6 + 7p) \)
\[
5(6 + 7p) = 5 \cdot 6 + 5 \cdot 7p = 30 + 35p
\]

#### Step 2: Add \( 10p \)
\[
10p + (30 + 35p) = 10p + 30 + 35p
\]

#### Step 3: Combine like terms
\[
10p + 35p = 45p
\]
\[
45p + 30
\]

#### Final Answer:
\[
\boxed{45p + 30}
\]

---

10. Simplify \( 5t^2 - 4t(2t + 4q) \)

#### Step 1: Distribute \( -4t \) across \( (2t + 4q) \)
\[
-4t(2t + 4q) = -4t \cdot 2t + (-4t) \cdot 4q
\]
\[
= -8t^2 - 16tq
\]

#### Step 2: Combine with \( 5t^2 \)
\[
5t^2 + (-8t^2 - 16tq) = 5t^2 - 8t^2 - 16tq
\]

#### Step 3: Combine like terms
\[
5t^2 - 8t^2 = -3t^2
\]
\[
-3t^2 - 16tq
\]

#### Final Answer:
\[
\boxed{-3t^2 - 16tq}
\]

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Final Answers:

1. \( \boxed{6t^2 + 9t} \)
2. \( \boxed{-16k^2} \)
3. \( \boxed{-2m^2 - 3m - 8} \)
4. \( \boxed{21x^2 - 27xy + 9x} \)
5. \( \boxed{-3w^2 + 5w - 5t} \)
6. \( \boxed{25w + 20wy} \)
7. \( \boxed{8pz - 12z^2 + 8p} \)
8. \( \boxed{-28n + 40p + 12} \)
9. \( \boxed{45p + 30} \)
10. \( \boxed{-3t^2 - 16tq} \)