Comprehensive worksheet designed to help students master the power of a product rule for exponents through varied practice problems.

Problem Analysis:

The worksheet focuses on the Power of a Product Rule in exponents. The rule states that for any nonzero numbers \(a\) and \(b\), and any integer \(n\):

\[
(a \cdot b)^n = a^n \cdot b^n
\]

We will solve each part of the problem step by step.

---

Part 1: Simplify Each Expression Using the Power of a Product Rule

#### (i) \((2x^3y^2)^4\)
Using the Power of a Product Rule:
\[
(2x^3y^2)^4 = 2^4 \cdot (x^3)^4 \cdot (y^2)^4
\]
Simplify each term:
- \(2^4 = 16\)
- \((x^3)^4 = x^{3 \cdot 4} = x^{12}\)
- \((y^2)^4 = y^{2 \cdot 4} = y^8\)

Thus:
\[
(2x^3y^2)^4 = 16x^{12}y^8
\]

#### (ii) \((3ab^2)^3\)
Using the Power of a Product Rule:
\[
(3ab^2)^3 = 3^3 \cdot a^3 \cdot (b^2)^3
\]
Simplify each term:
- \(3^3 = 27\)
- \(a^3 = a^3\)
- \((b^2)^3 = b^{2 \cdot 3} = b^6\)

Thus:
\[
(3ab^2)^3 = 27a^3b^6
\]

#### (iii) \((-5m^2n^3)^2\)
Using the Power of a Product Rule:
\[
(-5m^2n^3)^2 = (-5)^2 \cdot (m^2)^2 \cdot (n^3)^2
\]
Simplify each term:
- \((-5)^2 = 25\)
- \((m^2)^2 = m^{2 \cdot 2} = m^4\)
- \((n^3)^2 = n^{3 \cdot 2} = n^6\)

Thus:
\[
(-5m^2n^3)^2 = 25m^4n^6
\]

#### (iv) \((\frac{1}{2}xy^4)^3\)
Using the Power of a Product Rule:
\[
\left(\frac{1}{2}xy^4\right)^3 = \left(\frac{1}{2}\right)^3 \cdot x^3 \cdot (y^4)^3
\]
Simplify each term:
- \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\)
- \(x^3 = x^3\)
- \((y^4)^3 = y^{4 \cdot 3} = y^{12}\)

Thus:
\[
\left(\frac{1}{2}xy^4\right)^3 = \frac{1}{8}x^3y^{12}
\]

#### (v) \((4p^2q^3r)^2\)
Using the Power of a Product Rule:
\[
(4p^2q^3r)^2 = 4^2 \cdot (p^2)^2 \cdot (q^3)^2 \cdot r^2
\]
Simplify each term:
- \(4^2 = 16\)
- \((p^2)^2 = p^{2 \cdot 2} = p^4\)
- \((q^3)^2 = q^{3 \cdot 2} = q^6\)
- \(r^2 = r^2\)

Thus:
\[
(4p^2q^3r)^2 = 16p^4q^6r^2
\]

#### (vi) \((-2xyz^2)^3\)
Using the Power of a Product Rule:
\[
(-2xyz^2)^3 = (-2)^3 \cdot x^3 \cdot y^3 \cdot (z^2)^3
\]
Simplify each term:
- \((-2)^3 = -8\)
- \(x^3 = x^3\)
- \(y^3 = y^3\)
- \((z^2)^3 = z^{2 \cdot 3} = z^6\)

Thus:
\[
(-2xyz^2)^3 = -8x^3y^3z^6
\]

---

Part 2: Find the Value of \(t\)

#### (i) \((3t^2)^3 = 27t^6\)
Using the Power of a Product Rule:
\[
(3t^2)^3 = 3^3 \cdot (t^2)^3
\]
Simplify each term:
- \(3^3 = 27\)
- \((t^2)^3 = t^{2 \cdot 3} = t^6\)

Thus:
\[
(3t^2)^3 = 27t^6
\]
This matches the given expression, so \(t\) can be any real number. However, since no specific value is required, we conclude:
\[
t = \text{any real number}
\]

#### (ii) \((2t^3)^2 = 4t^6\)
Using the Power of a Product Rule:
\[
(2t^3)^2 = 2^2 \cdot (t^3)^2
\]
Simplify each term:
- \(2^2 = 4\)
- \((t^3)^2 = t^{3 \cdot 2} = t^6\)

Thus:
\[
(2t^3)^2 = 4t^6
\]
This matches the given expression, so \(t\) can be any real number. Again:
\[
t = \text{any real number}
\]

#### (iii) \((t^2 \cdot 3t)^3 = 27t^9\)
First, simplify inside the parentheses:
\[
t^2 \cdot 3t = 3t^3
\]
Now apply the power:
\[
(3t^3)^3 = 3^3 \cdot (t^3)^3
\]
Simplify each term:
- \(3^3 = 27\)
- \((t^3)^3 = t^{3 \cdot 3} = t^9\)

Thus:
\[
(3t^3)^3 = 27t^9
\]
This matches the given expression, so \(t\) can be any real number:
\[
t = \text{any real number}
\]

#### (iv) \((t^3 \cdot 2t^2)^2 = 4t^{10}\)
First, simplify inside the parentheses:
\[
t^3 \cdot 2t^2 = 2t^{3+2} = 2t^5
\]
Now apply the power:
\[
(2t^5)^2 = 2^2 \cdot (t^5)^2
\]
Simplify each term:
- \(2^2 = 4\)
- \((t^5)^2 = t^{5 \cdot 2} = t^{10}\)

Thus:
\[
(2t^5)^2 = 4t^{10}
\]
This matches the given expression, so \(t\) can be any real number:
\[
t = \text{any real number}
\]

#### (v) \((t^4 \cdot 3t^2)^3 = 27t^{18}\)
First, simplify inside the parentheses:
\[
t^4 \cdot 3t^2 = 3t^{4+2} = 3t^6
\]
Now apply the power:
\[
(3t^6)^3 = 3^3 \cdot (t^6)^3
\]
Simplify each term:
- \(3^3 = 27\)
- \((t^6)^3 = t^{6 \cdot 3} = t^{18}\)

Thus:
\[
(3t^6)^3 = 27t^{18}
\]
This matches the given expression, so \(t\) can be any real number:
\[
t = \text{any real number}
\]

#### (vi) \((t^2 \cdot 4t^3)^2 = 16t^{10}\)
First, simplify inside the parentheses:
\[
t^2 \cdot 4t^3 = 4t^{2+3} = 4t^5
\]
Now apply the power:
\[
(4t^5)^2 = 4^2 \cdot (t^5)^2
\]
Simplify each term:
- \(4^2 = 16\)
- \((t^5)^2 = t^{5 \cdot 2} = t^{10}\)

Thus:
\[
(4t^5)^2 = 16t^{10}
\]
This matches the given expression, so \(t\) can be any real number:
\[
t = \text{any real number}
\]

---

Part 3: Which of the Following Equals \((2x^3y^2)^4\)?

We already simplified \((2x^3y^2)^4\) in Part 1:
\[
(2x^3y^2)^4 = 16x^{12}y^8
\]

Now compare with the given options:
- (A) \(16x^4y^8\)
- (B) \(16x^{12}y^8\)
- (C) \(8x^{12}y^8\)
- (D) \(16x^3y^2\)

The correct option is:
\[
\boxed{B}
\]

---

Final Answers:

1. (i) \(16x^{12}y^8\)
(ii) \(27a^3b^6\)
(iii) \(25m^4n^6\)
(iv) \(\frac{1}{8}x^3y^{12}\)
(v) \(16p^4q^6r^2\)
(vi) \(-8x^3y^3z^6\)

2. (i) \(t = \text{any real number}\)
(ii) \(t = \text{any real number}\)
(iii) \(t = \text{any real number}\)
(iv) \(t = \text{any real number}\)
(v) \(t = \text{any real number}\)
(vi) \(t = \text{any real number}\)

3. \(\boxed{B}\)