Series of algebraic expressions demonstrating operations with fractions and variables.

Let's solve each expression step by step. We are given a series of algebraic expressions involving division and multiplication of rational expressions. The goal is to simplify each one.

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Expression 1:

$$
\frac{7x}{xy} \div \frac{6y}{3} \div \frac{4y}{14}
$$

Step 1: Simplify each fraction:

- $\frac{7x}{xy} = \frac{7}{y}$ (cancel $x$)
- $\frac{6y}{3} = 2y$
- $\frac{4y}{14} = \frac{2y}{7}$

Now substitute:

$$
\frac{7}{y} \div 2y \div \frac{2y}{7}
$$

Step 2: Convert divisions into multiplications:

Recall: $a \div b = a \times \frac{1}{b}$

So:
$$
\frac{7}{y} \div 2y = \frac{7}{y} \times \frac{1}{2y} = \frac{7}{2y^2}
$$

Then:
$$
\frac{7}{2y^2} \div \frac{2y}{7} = \frac{7}{2y^2} \times \frac{7}{2y} = \frac{49}{4y^3}
$$

✔ Answer for Expression 1: $\boxed{\frac{49}{4y^3}}$

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Expression 2:

$$
\frac{7x^2}{xy} \div \frac{6y}{3} \div \frac{4y}{14}
$$

Step 1: Simplify:

- $\frac{7x^2}{xy} = \frac{7x}{y}$
- $\frac{6y}{3} = 2y$
- $\frac{4y}{14} = \frac{2y}{7}$

Now:
$$
\frac{7x}{y} \div 2y \div \frac{2y}{7}
$$

Step 2: First division:
$$
\frac{7x}{y} \div 2y = \frac{7x}{y} \times \frac{1}{2y} = \frac{7x}{2y^2}
$$

Then:
$$
\frac{7x}{2y^2} \div \frac{2y}{7} = \frac{7x}{2y^2} \times \frac{7}{2y} = \frac{49x}{4y^3}
$$

✔ Answer for Expression 2: $\boxed{\frac{49x}{4y^3}}$

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Expression 3:

$$
\frac{7x^2}{xy} \div \left( \frac{6y}{3} \div \frac{4y}{14} \right)
$$

First, simplify inside the parentheses:

- $\frac{6y}{3} = 2y$
- $\frac{4y}{14} = \frac{2y}{7}$

So:
$$
2y \div \frac{2y}{7} = 2y \times \frac{7}{2y} = 7
$$

Now the whole expression becomes:
$$
\frac{7x^2}{xy} \div 7
$$

Simplify $\frac{7x^2}{xy} = \frac{7x}{y}$

Then:
$$
\frac{7x}{y} \div 7 = \frac{7x}{y} \times \frac{1}{7} = \frac{x}{y}
$$

✔ Answer for Expression 3: $\boxed{\frac{x}{y}}$

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Expression 4:

$$
\frac{7x^2}{xy} \div \left( \frac{6y}{3} \times \frac{4y}{14} \right)
$$

First, simplify inside the parentheses:

- $\frac{6y}{3} = 2y$
- $\frac{4y}{14} = \frac{2y}{7}$

Multiply:
$$
2y \times \frac{2y}{7} = \frac{4y^2}{7}
$$

Now:
$$
\frac{7x^2}{xy} \div \frac{4y^2}{7}
$$

Simplify $\frac{7x^2}{xy} = \frac{7x}{y}$

Now:
$$
\frac{7x}{y} \div \frac{4y^2}{7} = \frac{7x}{y} \times \frac{7}{4y^2} = \frac{49x}{4y^3}
$$

✔ Answer for Expression 4: $\boxed{\frac{49x}{4y^3}}$

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Expression 5:

$$
\frac{7x^2}{xy} \div \left( \frac{6y}{9} \times \frac{4y}{14} \right)
$$

First, simplify fractions:

- $\frac{6y}{9} = \frac{2y}{3}$
- $\frac{4y}{14} = \frac{2y}{7}$

Now multiply:
$$
\frac{2y}{3} \times \frac{2y}{7} = \frac{4y^2}{21}
$$

Now:
$$
\frac{7x^2}{xy} \div \frac{4y^2}{21}
$$

Simplify $\frac{7x^2}{xy} = \frac{7x}{y}$

Then:
$$
\frac{7x}{y} \div \frac{4y^2}{21} = \frac{7x}{y} \times \frac{21}{4y^2} = \frac{147x}{4y^3}
$$

✔ Answer for Expression 5: $\boxed{\frac{147x}{4y^3}}$

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✔ Final Answers:

1. $\frac{49}{4y^3}$
2. $\frac{49x}{4y^3}$
3. $\frac{x}{y}$
4. $\frac{49x}{4y^3}$
5. $\frac{147x}{4y^3}$

Let me know if you'd like these boxed or formatted differently!