Expansion of Algebraic Expressions worksheet - Free Printable

Problem: Simplifying Expressions with Brackets

The task involves simplifying algebraic expressions by distributing terms and combining like terms. Below is the step-by-step solution for each problem.

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#### 1. \( p^2q (m^2q + np) \)

Solution:
- Distribute \( p^2q \) to both terms inside the brackets:
\[
p^2q \cdot m^2q + p^2q \cdot np = p^2q \cdot m^2q + p^2q \cdot np
\]
- Simplify each term:
\[
p^2q \cdot m^2q = p^2m^2q^2
\]
\[
p^2q \cdot np = p^3qn
\]
- Combine the results:
\[
p^2m^2q^2 + p^3qn
\]

Answer:
\[
\boxed{p^2m^2q^2 + p^3qn}
\]

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#### 2. \( 3x^3 (4y - 7xy^2 + 8) \)

Solution:
- Distribute \( 3x^3 \) to each term inside the brackets:
\[
3x^3 \cdot 4y + 3x^3 \cdot (-7xy^2) + 3x^3 \cdot 8
\]
- Simplify each term:
\[
3x^3 \cdot 4y = 12x^3y
\]
\[
3x^3 \cdot (-7xy^2) = -21x^4y^2
\]
\[
3x^3 \cdot 8 = 24x^3
\]
- Combine the results:
\[
12x^3y - 21x^4y^2 + 24x^3
\]

Answer:
\[
\boxed{12x^3y - 21x^4y^2 + 24x^3}
\]

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#### 3. \( -5h^2 (6 - 2g^3 + g^2h^4) \)

Solution:
- Distribute \( -5h^2 \) to each term inside the brackets:
\[
-5h^2 \cdot 6 + (-5h^2) \cdot (-2g^3) + (-5h^2) \cdot g^2h^4
\]
- Simplify each term:
\[
-5h^2 \cdot 6 = -30h^2
\]
\[
(-5h^2) \cdot (-2g^3) = 10h^2g^3
\]
\[
(-5h^2) \cdot g^2h^4 = -5h^6g^2
\]
- Combine the results:
\[
-30h^2 + 10h^2g^3 - 5h^6g^2
\]

Answer:
\[
\boxed{-30h^2 + 10h^2g^3 - 5h^6g^2}
\]

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#### 4. \( 8ab (-2a + 3b^2 - 7b) \)

Solution:
- Distribute \( 8ab \) to each term inside the brackets:
\[
8ab \cdot (-2a) + 8ab \cdot 3b^2 + 8ab \cdot (-7b)
\]
- Simplify each term:
\[
8ab \cdot (-2a) = -16a^2b
\]
\[
8ab \cdot 3b^2 = 24ab^3
\]
\[
8ab \cdot (-7b) = -56ab^2
\]
- Combine the results:
\[
-16a^2b + 24ab^3 - 56ab^2
\]

Answer:
\[
\boxed{-16a^2b + 24ab^3 - 56ab^2}
\]

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#### 5. \( 3(x - 2y) + 4(y + 3x) \)

Solution:
- Distribute the constants to the terms inside the brackets:
\[
3 \cdot x + 3 \cdot (-2y) + 4 \cdot y + 4 \cdot 3x
\]
- Simplify each term:
\[
3x - 6y + 4y + 12x
\]
- Combine like terms:
\[
(3x + 12x) + (-6y + 4y) = 15x - 2y
\]

Answer:
\[
\boxed{15x - 2y}
\]

---

#### 6. \( 7m(m + 3n) - 6m(2n - m) \)

Solution:
- Distribute \( 7m \) and \( -6m \) to the terms inside their respective brackets:
\[
7m \cdot m + 7m \cdot 3n - 6m \cdot 2n - 6m \cdot (-m)
\]
- Simplify each term:
\[
7m^2 + 21mn - 12mn + 6m^2
\]
- Combine like terms:
\[
(7m^2 + 6m^2) + (21mn - 12mn) = 13m^2 + 9mn
\]

Answer:
\[
\boxed{13m^2 + 9mn}
\]

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

Solution:
- Distribute \( 9y \) and \( 2y \) to the terms inside their respective brackets:
\[
9y \cdot w + 9y \cdot (-x) + 9y \cdot 2y + 2y \cdot 7x + 2y \cdot (-3w) + 2y \cdot (-4y)
\]
- Simplify each term:
\[
9yw - 9yx + 18y^2 + 14yx - 6yw - 8y^2
\]
- Combine like terms:
\[
(9yw - 6yw) + (-9yx + 14yx) + (18y^2 - 8y^2) = 3yw + 5yx + 10y^2
\]

Answer:
\[
\boxed{3yw + 5yx + 10y^2}
\]

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#### 8. \( 2(r - s) - 4(s - 5r) - (3r - 6s) \)

Solution:
- Distribute the constants to the terms inside the brackets:
\[
2 \cdot r + 2 \cdot (-s) - 4 \cdot s - 4 \cdot (-5r) - 3r + 6s
\]
- Simplify each term:
\[
2r - 2s - 4s + 20r - 3r + 6s
\]
- Combine like terms:
\[
(2r + 20r - 3r) + (-2s - 4s + 6s) = 19r + 0s = 19r
\]

Answer:
\[
\boxed{19r}
\]

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#### 9. \( 19g(h - 2g) - 15g(2h - 3g) + 2g(2h - 5g) \)

Solution:
- Distribute \( 19g \), \( -15g \), and \( 2g \) to the terms inside their respective brackets:
\[
19g \cdot h + 19g \cdot (-2g) - 15g \cdot 2h - 15g \cdot (-3g) + 2g \cdot 2h + 2g \cdot (-5g)
\]
- Simplify each term:
\[
19gh - 38g^2 - 30gh + 45g^2 + 4gh - 10g^2
\]
- Combine like terms:
\[
(19gh - 30gh + 4gh) + (-38g^2 + 45g^2 - 10g^2) = 3gh - 3g^2
\]

Answer:
\[
\boxed{3gh - 3g^2}
\]

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#### 10. \( 12m(4n - p) - 5m(2n + 3p) \)

Solution:
- Distribute \( 12m \) and \( -5m \) to the terms inside their respective brackets:
\[
12m \cdot 4n + 12m \cdot (-p) - 5m \cdot 2n - 5m \cdot 3p
\]
- Simplify each term:
\[
48mn - 12mp - 10mn - 15mp
\]
- Combine like terms:
\[
(48mn - 10mn) + (-12mp - 15mp) = 38mn - 27mp
\]

Answer:
\[
\boxed{38mn - 27mp}
\]

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#### 11. \( 14n(em - 3n + 2) - 3n(10 - 16n + 3em) - 7n + 2n^2 \)

Solution:
- Distribute \( 14n \) and \( -3n \) to the terms inside their respective brackets:
\[
14n \cdot em + 14n \cdot (-3n) + 14n \cdot 2 - 3n \cdot 10 - 3n \cdot (-16n) - 3n \cdot 3em - 7n + 2n^2
\]
- Simplify each term:
\[
14nem - 42n^2 + 28n - 30n + 48n^2 - 9nem - 7n + 2n^2
\]
- Combine like terms:
\[
(14nem - 9nem) + (-42n^2 + 48n^2 + 2n^2) + (28n - 30n - 7n) = 5nem + 8n^2 - 9n
\]

Answer:
\[
\boxed{5nem + 8n^2 - 9n}
\]

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Final Answers:

1. \(\boxed{p^2m^2q^2 + p^3qn}\)
2. \(\boxed{12x^3y - 21x^4y^2 + 24x^3}\)
3. \(\boxed{-30h^2 + 10h^2g^3 - 5h^6g^2}\)
4. \(\boxed{-16a^2b + 24ab^3 - 56ab^2}\)
5. \(\boxed{15x - 2y}\)
6. \(\boxed{13m^2 + 9mn}\)
7. \(\boxed{3yw + 5yx + 10y^2}\)
8. \(\boxed{19r}\)
9. \(\boxed{3gh - 3g^2}\)
10. \(\boxed{38mn - 27mp}\)
11. \(\boxed{5nem + 8n^2 - 9n}\)