Let's analyze and solve each problem step-by-step from the image.

1. Problem 1

$\frac{(a + b)c}{d} = f$ Find $b$.

Solution:

1. Multiply both sides by $d$: $(a + b)c = f \cdot d$

2. Divide both sides by $c$: $a + b = \frac{f \cdot d}{c}$

3. Subtract $a$ from both sides to solve for $b$: $b = \frac{f \cdot d}{c} - a$

2. Problem 2

$\frac{a^5 \cdot (b + c) \cdot d}{e} = ?$ Find $a$.

Solution:

Without a specific equation for $a$, the expression given can be simplified or rearranged but cannot be solved for $a$ explicitly unless more context or values are provided. For now, the expression remains as given:

$\text{Expression } = \frac{a^5 \cdot (b + c) \cdot d}{e}$

3. Problem 3

Simplify: $a^{\frac{1}{3}} \times a^{\frac{3}{5}} \times a^{\frac{7}{2}}$

Solution:

1. Use the property of exponents: $a^m \times a^n = a^{m+n}$.

2. Add the exponents: $\frac{1}{3} + \frac{3}{5} + \frac{7}{2}$

3. Find a common denominator (which is 30): $\frac{1}{3} = \frac{10}{30}, \quad \frac{3}{5} = \frac{18}{30}, \quad \frac{7}{2} = \frac{105}{30}$

4. Adding them together: $\frac{10}{30} + \frac{18}{30} + \frac{105}{30} = \frac{133}{30}$

5. So, the expression simplifies to: $a^{\frac{133}{30}}$

Let me know if you'd like more detailed explanations or have any other questions!

Relative Questions:

1. How can we solve for other variables in the first equation if given their values?
2. What are the steps to simplify expressions with multiple variables and exponents?
3. How can you factor expressions like the ones in problem 2?
4. What is the significance of finding common denominators in adding fractions?
5. How can we rewrite exponential expressions in radical form?

Tip: Always check for common factors in the numerator and denominator when simplifying rational expressions.