Simplifying expressions with exponents worksheet for algebra practice.

Let's solve each expression step by step using the distributive property and rules of exponents.

We will simplify each expression by distributing the term outside the parentheses to each term inside.

---

1. $ 3r(r^2 + 4) $

Distribute $ 3r $:
$$
3r \cdot r^2 = 3r^3,\quad 3r \cdot 4 = 12r
$$
Answer: $ 3r^3 + 12r $

---

2. $ 2x(-8x^2 + 2) $

Distribute $ 2x $:
$$
2x \cdot (-8x^2) = -16x^3,\quad 2x \cdot 2 = 4x
$$
Answer: $ -16x^3 + 4x $

---

3. $ 2y(-3y - 9) $

Distribute $ 2y $:
$$
2y \cdot (-3y) = -6y^2,\quad 2y \cdot (-9) = -18y
$$
Answer: $ -6y^2 - 18y $

---

4. $ -2z(-6z - 1) $

Distribute $ -2z $:
$$
-2z \cdot (-6z) = 12z^2,\quad -2z \cdot (-1) = 2z
$$
Answer: $ 12z^2 + 2z $

---

5. $ 4q(-4 + 3q) $

Distribute $ 4q $:
$$
4q \cdot (-4) = -16q,\quad 4q \cdot 3q = 12q^2
$$
Answer: $ 12q^2 - 16q $

---

6. $ -2d(-2d^2 + 5d) $

Distribute $ -2d $:
$$
-2d \cdot (-2d^2) = 4d^3,\quad -2d \cdot 5d = -10d^2
$$
Answer: $ 4d^3 - 10d^2 $

---

7. $ -v(-4v + 4v^2) $

Distribute $ -v $:
$$
-v \cdot (-4v) = 4v^2,\quad -v \cdot 4v^2 = -4v^3
$$
Answer: $ 4v^2 - 4v^3 $
*(or written in standard form: $ -4v^3 + 4v^2 $)*

---

8. $ 2a(-6a^2 + 6c) $

Distribute $ 2a $:
$$
2a \cdot (-6a^2) = -12a^3,\quad 2a \cdot 6c = 12ac
$$
Answer: $ -12a^3 + 12ac $

---

9. $ -4k(-5 - k) $

Distribute $ -4k $:
$$
-4k \cdot (-5) = 20k,\quad -4k \cdot (-k) = 4k^2
$$
Answer: $ 4k^2 + 20k $

---

10. $ 3r(6 + 6r^2) $

Distribute $ 3r $:
$$
3r \cdot 6 = 18r,\quad 3r \cdot 6r^2 = 18r^3
$$
Answer: $ 18r^3 + 18r $

---

11. $ -d(-3d + 4) $

Distribute $ -d $:
$$
-d \cdot (-3d) = 3d^2,\quad -d \cdot 4 = -4d
$$
Answer: $ 3d^2 - 4d $

---

12. $ -2g(2g^2 + 4g) $

Distribute $ -2g $:
$$
-2g \cdot 2g^2 = -4g^3,\quad -2g \cdot 4g = -8g^2
$$
Answer: $ -4g^3 - 8g^2 $

---

13. $ 3x^2(5 - 2x) $

Distribute $ 3x^2 $:
$$
3x^2 \cdot 5 = 15x^2,\quad 3x^2 \cdot (-2x) = -6x^3
$$
Answer: $ -6x^3 + 15x^2 $

---

14. $ 2h(6h + 5) $

Distribute $ 2h $:
$$
2h \cdot 6h = 12h^2,\quad 2h \cdot 5 = 10h
$$
Answer: $ 12h^2 + 10h $

---

15. $ 6e(5v - 5e) $

Distribute $ 6e $:
$$
6e \cdot 5v = 30ev,\quad 6e \cdot (-5e) = -30e^2
$$
Answer: $ 30ev - 30e^2 $

---

16. $ 4b(-3b + 9b) $

First, simplify inside the parentheses:
$$
-3b + 9b = 6b
$$
Now: $ 4b \cdot 6b = 24b^2 $
Answer: $ 24b^2 $

---

17. $ 5(3q + 5q) $

Simplify inside:
$$
3q + 5q = 8q
$$
Then: $ 5 \cdot 8q = 40q $
Answer: $ 40q $

---

18. $ 2n(3n^2 + 5) $

Distribute $ 2n $:
$$
2n \cdot 3n^2 = 6n^3,\quad 2n \cdot 5 = 10n
$$
Answer: $ 6n^3 + 10n $

---

19. $ -3m^2(-1 + m) $

Distribute $ -3m^2 $:
$$
-3m^2 \cdot (-1) = 3m^2,\quad -3m^2 \cdot m = -3m^3
$$
Answer: $ -3m^3 + 3m^2 $

---

20. $ 2r(2r^4 + 5) $

Distribute $ 2r $:
$$
2r \cdot 2r^4 = 4r^5,\quad 2r \cdot 5 = 10r
$$
Answer: $ 4r^5 + 10r $

---

21. $ 8v(-2v^2 + 4f) $

Distribute $ 8v $:
$$
8v \cdot (-2v^2) = -16v^3,\quad 8v \cdot 4f = 32vf
$$
Answer: $ -16v^3 + 32vf $

---

## ✔ Final Answers (Listed):

1. $ 3r^3 + 12r $
2. $ -16x^3 + 4x $
3. $ -6y^2 - 18y $
4. $ 12z^2 + 2z $
5. $ 12q^2 - 16q $
6. $ 4d^3 - 10d^2 $
7. $ -4v^3 + 4v^2 $
8. $ -12a^3 + 12ac $
9. $ 4k^2 + 20k $
10. $ 18r^3 + 18r $
11. $ 3d^2 - 4d $
12. $ -4g^3 - 8g^2 $
13. $ -6x^3 + 15x^2 $
14. $ 12h^2 + 10h $
15. $ 30ev - 30e^2 $
16. $ 24b^2 $
17. $ 40q $
18. $ 6n^3 + 10n $
19. $ -3m^3 + 3m^2 $
20. $ 4r^5 + 10r $
21. $ -16v^3 + 32vf $

---

🔍 Key Concepts Used:

- Distributive Property: $ a(b + c) = ab + ac $
- Exponent Rules: When multiplying like bases, add exponents: $ x^a \cdot x^b = x^{a+b} $
- Sign Rules: Negative times negative = positive; negative times positive = negative

Let me know if you'd like this as a printable PDF or with steps highlighted!