Math worksheet focusing on simplifying expressions using the laws of exponents.

Problem: Simplify the following expressions using the laws of exponents.

The laws of exponents provided are:
1. \( a^m \cdot a^n = a^{m+n} \)
2. \( \frac{a^m}{a^n} = a^{m-n} \)
3. \( (a^m)^n = a^{m \cdot n} \)
4. \( (a^m b^n)^p = a^{mp} b^{np} \)
5. \( \left( \frac{a^m}{b^n} \right)^p = \frac{a^{mp}}{b^{np}} \)

Let's solve each expression step by step.

---

1. \( a^3 \cdot a^5 \)

Using the law: \( a^m \cdot a^n = a^{m+n} \)

\[
a^3 \cdot a^5 = a^{3+5} = a^8
\]

Answer: \( a^8 \)

---

2. \( a^3 b \cdot ab^6 \)

First, group the like terms:

\[
a^3 b \cdot ab^6 = (a^3 \cdot a) \cdot (b \cdot b^6)
\]

Using the law \( a^m \cdot a^n = a^{m+n} \):

\[
a^3 \cdot a = a^{3+1} = a^4
\]
\[
b \cdot b^6 = b^{1+6} = b^7
\]

So,

\[
a^3 b \cdot ab^6 = a^4 b^7
\]

Answer: \( a^4 b^7 \)

---

3. \( (2b^5)^3 \)

Using the law \( (ab)^n = a^n b^n \):

\[
(2b^5)^3 = 2^3 \cdot (b^5)^3
\]

Calculate \( 2^3 \):

\[
2^3 = 8
\]

Using the law \( (a^m)^n = a^{m \cdot n} \):

\[
(b^5)^3 = b^{5 \cdot 3} = b^{15}
\]

So,

\[
(2b^5)^3 = 8b^{15}
\]

Answer: \( 8b^{15} \)

---

4. \( (a^4 b^2)^3 \)

Using the law \( (a^m b^n)^p = a^{mp} b^{np} \):

\[
(a^4 b^2)^3 = (a^4)^3 \cdot (b^2)^3
\]

Using the law \( (a^m)^n = a^{m \cdot n} \):

\[
(a^4)^3 = a^{4 \cdot 3} = a^{12}
\]
\[
(b^2)^3 = b^{2 \cdot 3} = b^6
\]

So,

\[
(a^4 b^2)^3 = a^{12} b^6
\]

Answer: \( a^{12} b^6 \)

---

5. \( (a^2)^3 (a^2)^2 \)

Using the law \( (a^m)^n = a^{m \cdot n} \):

\[
(a^2)^3 = a^{2 \cdot 3} = a^6
\]
\[
(a^2)^2 = a^{2 \cdot 2} = a^4
\]

Now, multiply the results using \( a^m \cdot a^n = a^{m+n} \):

\[
a^6 \cdot a^4 = a^{6+4} = a^{10}
\]

Answer: \( a^{10} \)

---

6. \( (3b^2)(2b) \)

First, multiply the coefficients and then the variables:

\[
(3b^2)(2b) = (3 \cdot 2) \cdot (b^2 \cdot b)
\]

Calculate the coefficients:

\[
3 \cdot 2 = 6
\]

Using the law \( b^m \cdot b^n = b^{m+n} \):

\[
b^2 \cdot b = b^{2+1} = b^3
\]

So,

\[
(3b^2)(2b) = 6b^3
\]

Answer: \( 6b^3 \)

---

7. \( \frac{a^7}{a^3} \)

Using the law \( \frac{a^m}{a^n} = a^{m-n} \):

\[
\frac{a^7}{a^3} = a^{7-3} = a^4
\]

Answer: \( a^4 \)

---

8. \( \left( \frac{a^3 b^5}{ab} \right)^2 \)

First, simplify the fraction inside the parentheses:

\[
\frac{a^3 b^5}{ab} = \frac{a^3}{a} \cdot \frac{b^5}{b}
\]

Using the law \( \frac{a^m}{a^n} = a^{m-n} \):

\[
\frac{a^3}{a} = a^{3-1} = a^2
\]
\[
\frac{b^5}{b} = b^{5-1} = b^4
\]

So,

\[
\frac{a^3 b^5}{ab} = a^2 b^4
\]

Now, raise the simplified expression to the power of 2:

\[
\left( a^2 b^4 \right)^2 = (a^2)^2 \cdot (b^4)^2
\]

Using the law \( (a^m)^n = a^{m \cdot n} \):

\[
(a^2)^2 = a^{2 \cdot 2} = a^4
\]
\[
(b^4)^2 = b^{4 \cdot 2} = b^8
\]

So,

\[
\left( \frac{a^3 b^5}{ab} \right)^2 = a^4 b^8
\]

Answer: \( a^4 b^8 \)

---

9. \( \frac{(4a^4)^2}{(2a)^3} \)

First, simplify the numerator and the denominator separately.

#### Numerator: \( (4a^4)^2 \)

Using the law \( (ab)^n = a^n b^n \):

\[
(4a^4)^2 = 4^2 \cdot (a^4)^2
\]

Calculate \( 4^2 \):

\[
4^2 = 16
\]

Using the law \( (a^m)^n = a^{m \cdot n} \):

\[
(a^4)^2 = a^{4 \cdot 2} = a^8
\]

So,

\[
(4a^4)^2 = 16a^8
\]

#### Denominator: \( (2a)^3 \)

Using the law \( (ab)^n = a^n b^n \):

\[
(2a)^3 = 2^3 \cdot a^3
\]

Calculate \( 2^3 \):

\[
2^3 = 8
\]

So,

\[
(2a)^3 = 8a^3
\]

Now, divide the simplified numerator by the simplified denominator:

\[
\frac{(4a^4)^2}{(2a)^3} = \frac{16a^8}{8a^3}
\]

Simplify the coefficients and the variables separately:

\[
\frac{16}{8} = 2
\]

Using the law \( \frac{a^m}{a^n} = a^{m-n} \):

\[
\frac{a^8}{a^3} = a^{8-3} = a^5
\]

So,

\[
\frac{(4a^4)^2}{(2a)^3} = 2a^5
\]

Answer: \( 2a^5 \)

---

10. \( (5a^2 bc^4)^0 \)

Any non-zero number raised to the power of 0 is 1:

\[
(5a^2 bc^4)^0 = 1
\]

Answer: \( 1 \)

---

Final Answers:

1. \( a^8 \)
2. \( a^4 b^7 \)
3. \( 8b^{15} \)
4. \( a^{12} b^6 \)
5. \( a^{10} \)
6. \( 6b^3 \)
7. \( a^4 \)
8. \( a^4 b^8 \)
9. \( 2a^5 \)
10. \( 1 \)

\boxed{a^8, a^4 b^7, 8b^{15}, a^{12} b^6, a^{10}, 6b^3, a^4, a^4 b^8, 2a^5, 1}