Algebra 1 Worksheets | Exponents Worksheets - Free Printable

Problem: Simplify the given expressions. Your answer should contain only positive exponents.

We will solve each problem step by step using the rules of exponents:
1. Power of a product rule: $(ab)^n = a^n \cdot b^n$
2. Power of a power rule: $(a^m)^n = a^{m \cdot n}$
3. Product of powers rule: $a^m \cdot a^n = a^{m+n}$
4. Quotient of powers rule: $\frac{a^m}{a^n} = a^{m-n}$

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#### Problem 1: $(6w^5)^3$

- Apply the power of a product rule: $(ab)^n = a^n \cdot b^n$
- Here, $a = 6$ and $b = w^5$, so:
$$
(6w^5)^3 = 6^3 \cdot (w^5)^3
$$
- Simplify $6^3$ and apply the power of a power rule to $(w^5)^3$:
$$
6^3 = 216, \quad (w^5)^3 = w^{5 \cdot 3} = w^{15}
$$
- Combine the results:
$$
(6w^5)^3 = 216w^{15}
$$

Answer:
$$
\boxed{216w^{15}}
$$

---

#### Problem 2: $(2h \cdot 3h^3)^2$

- First, simplify inside the parentheses:
$$
2h \cdot 3h^3 = (2 \cdot 3) \cdot (h \cdot h^3) = 6h^4
$$
- Now apply the power of a product rule:
$$
(6h^4)^2 = 6^2 \cdot (h^4)^2
$$
- Simplify $6^2$ and apply the power of a power rule to $(h^4)^2$:
$$
6^2 = 36, \quad (h^4)^2 = h^{4 \cdot 2} = h^8
$$
- Combine the results:
$$
(6h^4)^2 = 36h^8
$$

Answer:
$$
\boxed{36h^8}
$$

---

#### Problem 3: $(4g \cdot 3g^3 \cdot g^2)^2$

- First, simplify inside the parentheses:
$$
4g \cdot 3g^3 \cdot g^2 = (4 \cdot 3) \cdot (g \cdot g^3 \cdot g^2) = 12g^{1+3+2} = 12g^6
$$
- Now apply the power of a product rule:
$$
(12g^6)^2 = 12^2 \cdot (g^6)^2
$$
- Simplify $12^2$ and apply the power of a power rule to $(g^6)^2$:
$$
12^2 = 144, \quad (g^6)^2 = g^{6 \cdot 2} = g^{12}
$$
- Combine the results:
$$
(12g^6)^2 = 144g^{12}
$$

Answer:
$$
\boxed{144g^{12}}
$$

---

#### Problem 4: $(4hr^2)^2$

- Apply the power of a product rule:
$$
(4hr^2)^2 = 4^2 \cdot h^2 \cdot (r^2)^2
$$
- Simplify $4^2$ and apply the power of a power rule to $(r^2)^2$:
$$
4^2 = 16, \quad (r^2)^2 = r^{2 \cdot 2} = r^4
$$
- Combine the results:
$$
(4hr^2)^2 = 16h^2r^4
$$

Answer:
$$
\boxed{16h^2r^4}
$$

---

#### Problem 5: $(4w^3 \cdot w^2 \cdot 3w)^2$

- First, simplify inside the parentheses:
$$
4w^3 \cdot w^2 \cdot 3w = (4 \cdot 3) \cdot (w^3 \cdot w^2 \cdot w) = 12w^{3+2+1} = 12w^6
$$
- Now apply the power of a product rule:
$$
(12w^6)^2 = 12^2 \cdot (w^6)^2
$$
- Simplify $12^2$ and apply the power of a power rule to $(w^6)^2$:
$$
12^2 = 144, \quad (w^6)^2 = w^{6 \cdot 2} = w^{12}
$$
- Combine the results:
$$
(12w^6)^2 = 144w^{12}
$$

Answer:
$$
\boxed{144w^{12}}
$$

---

#### Problem 6: $(4z^3 \cdot 5x \cdot z^2)^2$

- First, simplify inside the parentheses:
$$
4z^3 \cdot 5x \cdot z^2 = (4 \cdot 5) \cdot (z^3 \cdot z^2) \cdot x = 20z^{3+2} \cdot x = 20z^5x
$$
- Now apply the power of a product rule:
$$
(20z^5x)^2 = 20^2 \cdot (z^5)^2 \cdot x^2
$$
- Simplify $20^2$ and apply the power of a power rule to $(z^5)^2$:
$$
20^2 = 400, \quad (z^5)^2 = z^{5 \cdot 2} = z^{10}
$$
- Combine the results:
$$
(20z^5x)^2 = 400z^{10}x^2
$$

Answer:
$$
\boxed{400z^{10}x^2}
$$

---

#### Problem 7: $(2d^2y)^2$

- Apply the power of a product rule:
$$
(2d^2y)^2 = 2^2 \cdot (d^2)^2 \cdot y^2
$$
- Simplify $2^2$ and apply the power of a power rule to $(d^2)^2$:
$$
2^2 = 4, \quad (d^2)^2 = d^{2 \cdot 2} = d^4
$$
- Combine the results:
$$
(2d^2y)^2 = 4d^4y^2
$$

Answer:
$$
\boxed{4d^4y^2}
$$

---

#### Problem 8: $(4c^2 \cdot c^3)^2$

- First, simplify inside the parentheses:
$$
4c^2 \cdot c^3 = 4 \cdot (c^2 \cdot c^3) = 4c^{2+3} = 4c^5
$$
- Now apply the power of a product rule:
$$
(4c^5)^2 = 4^2 \cdot (c^5)^2
$$
- Simplify $4^2$ and apply the power of a power rule to $(c^5)^2$:
$$
4^2 = 16, \quad (c^5)^2 = c^{5 \cdot 2} = c^{10}
$$
- Combine the results:
$$
(4c^5)^2 = 16c^{10}
$$

Answer:
$$
\boxed{16c^{10}}
$$

---

#### Problem 9: $(g \cdot 2g^2)^3$

- First, simplify inside the parentheses:
$$
g \cdot 2g^2 = (1 \cdot 2) \cdot (g \cdot g^2) = 2g^{1+2} = 2g^3
$$
- Now apply the power of a product rule:
$$
(2g^3)^3 = 2^3 \cdot (g^3)^3
$$
- Simplify $2^3$ and apply the power of a power rule to $(g^3)^3$:
$$
2^3 = 8, \quad (g^3)^3 = g^{3 \cdot 3} = g^9
$$
- Combine the results:
$$
(2g^3)^3 = 8g^9
$$

Answer:
$$
\boxed{8g^9}
$$

---

#### Problem 10: $(4n^2 \cdot n)^3$

- First, simplify inside the parentheses:
$$
4n^2 \cdot n = 4 \cdot (n^2 \cdot n) = 4n^{2+1} = 4n^3
$$
- Now apply the power of a product rule:
$$
(4n^3)^3 = 4^3 \cdot (n^3)^3
$$
- Simplify $4^3$ and apply the power of a power rule to $(n^3)^3$:
$$
4^3 = 64, \quad (n^3)^3 = n^{3 \cdot 3} = n^9
$$
- Combine the results:
$$
(4n^3)^3 = 64n^9
$$

Answer:
$$
\boxed{64n^9}
$$

---

#### Problem 11: $(s \cdot 3s^3 \cdot s^3)^2$

- First, simplify inside the parentheses:
$$
s \cdot 3s^3 \cdot s^3 = (1 \cdot 3) \cdot (s \cdot s^3 \cdot s^3) = 3s^{1+3+3} = 3s^7
$$
- Now apply the power of a product rule:
$$
(3s^7)^2 = 3^2 \cdot (s^7)^2
$$
- Simplify $3^2$ and apply the power of a power rule to $(s^7)^2$:
$$
3^2 = 9, \quad (s^7)^2 = s^{7 \cdot 2} = s^{14}
$$
- Combine the results:
$$
(3s^7)^2 = 9s^{14}
$$

Answer:
$$
\boxed{9s^{14}}
$$

---

#### Problem 12: $(3a^2 \cdot 4a)^3$

- First, simplify inside the parentheses:
$$
3a^2 \cdot 4a = (3 \cdot 4) \cdot (a^2 \cdot a) = 12a^{2+1} = 12a^3
$$
- Now apply the power of a product rule:
$$
(12a^3)^3 = 12^3 \cdot (a^3)^3
$$
- Simplify $12^3$ and apply the power of a power rule to $(a^3)^3$:
$$
12^3 = 1728, \quad (a^3)^3 = a^{3 \cdot 3} = a^9
$$
- Combine the results:
$$
(12a^3)^3 = 1728a^9
$$

Answer:
$$
\boxed{1728a^9}
$$

---

#### Problem 13: $(2z^2 \cdot 3z^3 \cdot z)^2$

- First, simplify inside the parentheses:
$$
2z^2 \cdot 3z^3 \cdot z = (2 \cdot 3) \cdot (z^2 \cdot z^3 \cdot z) = 6z^{2+3+1} = 6z^6
$$
- Now apply the power of a product rule:
$$
(6z^6)^2 = 6^2 \cdot (z^6)^2
$$
- Simplify $6^2$ and apply the power of a power rule to $(z^6)^2$:
$$
6^2 = 36, \quad (z^6)^2 = z^{6 \cdot 2} = z^{12}
$$
- Combine the results:
$$
(6z^6)^2 = 36z^{12}
$$

Answer:
$$
\boxed{36z^{12}}
$$

---

#### Problem 14: $(3y^2 \cdot y \cdot 4)^3$

- First, simplify inside the parentheses:
$$
3y^2 \cdot y \cdot 4 = (3 \cdot 4) \cdot (y^2 \cdot y) = 12y^{2+1} = 12y^3
$$
- Now apply the power of a product rule:
$$
(12y^3)^3 = 12^3 \cdot (y^3)^3
$$
- Simplify $12^3$ and apply the power of a power rule to $(y^3)^3$:
$$
12^3 = 1728, \quad (y^3)^3 = y^{3 \cdot 3} = y^9
$$
- Combine the results:
$$
(12y^3)^3 = 1728y^9
$$

Answer:
$$
\boxed{1728y^9}
$$

---

Final Answers:

1. $\boxed{216w^{15}}$
2. $\boxed{36h^8}$
3. $\boxed{144g^{12}}$
4. $\boxed{16h^2r^4}$
5. $\boxed{144w^{12}}$
6. $\boxed{400z^{10}x^2}$
7. $\boxed{4d^4y^2}$
8. $\boxed{16c^{10}}$
9. $\boxed{8g^9}$
10. $\boxed{64n^9}$
11. $\boxed{9s^{14}}$
12. $\boxed{1728a^9}$
13. $\boxed{36z^{12}}$
14. $\boxed{1728y^9}$