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Let's go through the problems one by one, evaluating each expression for \( s = 4 \) and \( t = 8 \).

### 25. \((s + t)^3\)
\[
(s + t)^3 = (4 + 8)^3 = 12^3 = 1728
\]
Answer: 1728

### 26. \(s^4 + t^2 + \frac{s}{2}\)
\[
s^4 + t^2 + \frac{s}{2} = 4^4 + 8^2 + \frac{4}{2} = 256 + 64 + 2 = 322
\]
Answer: 322

### 27. \(\frac{(st)^2}{(st^2)}\)
First, calculate \(st\) and \(st^2\):
\[
st = 4 \times 8 = 32, \quad st^2 = 4 \times 8^2 = 4 \times 64 = 256
\]
Now, calculate the expression:
\[
\frac{(st)^2}{(st^2)} = \frac{32^2}{256} = \frac{1024}{256} = 4
\]
Answer: 4

### 28. \(3st^2 \div (st) + 6\)
Calculate \(st^2\) and \(st\):
\[
st^2 = 256, \quad st = 32
\]
Now, compute the expression:
\[
3st^2 \div (st) + 6 = \frac{3 \times 256}{32} + 6 = 24 + 6 = 30
\]
Answer: 30

### 29. \((t - s)^5\)
\[
(t - s)^5 = (8 - 4)^5 = 4^5 = 1024
\]
Answer: 1024

### 30. \((2s)^2 t\)
\[
(2s)^2 t = (2 \times 4)^2 \times 8 = 8^2 \times 8 = 64 \times 8 = 512
\]
Answer: 512

### 31. \(2st^2 - s^3\)
\[
2st^2 - s^3 = 2 \times 256 - 4^3 = 512 - 64 = 448
\]
Answer: 448

### 32. \(2s^2 - t^3 \div 16\)
\[
2s^2 - \frac{t^3}{16} = 2 \times 4^2 - \frac{8^3}{16} = 2 \times 16 - \frac{512}{16} = 32 - 32 = 0
\]
Answer: 0

### 33. \(\frac{(3s)^3 t + t}{s}\)
First, compute \((3s)^3\) and the rest:
\[
(3s)^3 = (3 \times 4)^3 = 12^3 = 1728, \quad \frac{(3s)^3 t + t}{s} = \frac{1728 \times 8 + 8}{4} = \frac{13824 + 8}{4} = \frac{13832}{4} = 3458
\]
Answer: 3458

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Would you like more details on any of these steps?

### Related Questions:
1. How does the distributive property apply in expanding expressions like \( (s + t)^3 \)?
2. What happens to the expression when \( s \) or \( t \) changes values drastically?
3. Can you factor any of these expressions for different values of \( s \) and \( t \)?
4. How do powers affect the magnitude of results in expressions involving higher exponents?
5. How does the order of operations (PEMDAS) influence the calculation of complex expressions?

### Tip:
Always remember to apply the correct order of operations when working through expressions, especially when multiple operations like exponentiation, multiplication, and addition are involved.

Evaluate each expression for s = 4 and t = 8.

Solve algebraic expressions including exponents, multiplication, and division for s = 4 and t = 8. Learn to apply key concepts like exponentiation and the distributive property. Suitable for students in Grades 8-10.

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