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Let's go through each of the problems step-by-step to help you solve them.

### 1. Simplify the expression: 
\[
\left(\frac{pq^{\frac{1}{2}} r^3}{r^3}\right)^4
\]

Simplification steps:
1. Simplify the fraction inside the parentheses:
   \[
   \frac{pq^{\frac{1}{2}} r^3}{r^3} = pq^{\frac{1}{2}}
   \]
   since \(r^3\) cancels out.

2. Raise to the power of 4:
   \[
   (pq^{\frac{1}{2}})^4 = p^4 (q^{\frac{1}{2}})^4 = p^4 q^2.
   \]

So, the simplified form is:
\[
p^4 q^2.
\]

### 2. Simplify: 
\[
(2a^{-1} b)^3
\]

Simplification steps:
1. Distribute the power of 3:
   \[
   (2a^{-1} b)^3 = 2^3 (a^{-1})^3 b^3 = 8 a^{-3} b^3.
   \]

So, the simplified form is:
\[
8 a^{-3} b^3.
\]

### 3. Function Evaluation: 
\[
f(6) = 3 \cdot 2^x
\]

If \(x = 6\):
\[
f(6) = 3 \cdot 2^6 = 3 \cdot 64 = 192.
\]

### 4. Given \(A = 3\), \(b = 4\), and \(c = 1\), find the value of:
\[
\frac{a^6 b^{-2} c^b}{(bc)^b}.
\]

Substitute values:
1. \(a = A = 3\), \(b = 4\), \(c = 1\).
2. Substitute into the expression:
   \[
   \frac{3^6 \cdot 4^{-2} \cdot 1^4}{(4 \cdot 1)^4} = \frac{729 \cdot \frac{1}{16} \cdot 1}{4^4} = \frac{729}{16 \cdot 256}.
   \]

This needs further simplification:
\[
\frac{729}{4096}.
\]

### 5. Simplify: 
\[
3x^{2x+6} = 2y^{2x+3}.
\]

Since no specific simplification rule is specified, the expression remains as stated.

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### Do you want details or have any questions?

Here are some related questions:

1. How do you simplify expressions with fractional exponents?
2. What are the rules for distributing exponents over multiplication?
3. How do you evaluate a function with an exponent?
4. What is the procedure to simplify complex fractions with variables?
5. How can expressions involving different bases with exponents be equated?

**Tip:** When simplifying expressions with exponents, remember to apply the power rules carefully to each term individually.

Simplify mathematical expressions, including exponents, function evaluation, and algebraic expressions involving variables.

Learn how to simplify mathematical expressions involving exponents, algebraic terms, and functions. Includes step-by-step explanations and examples suitable for high school students.

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