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Let's go through each part of the problem step by step.

### (d) \(\left(\frac{x}{y}\right)^{-2}\)

To simplify \(\left(\frac{x}{y}\right)^{-2}\), apply the negative exponent rule:
\[
\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
\]
Thus:
\[
\left(\frac{x}{y}\right)^{-2} = \left(\frac{y}{x}\right)^2 = \frac{y^2}{x^2}
\]

So, the simplified form is:
\[
\frac{y^2}{x^2}
\]

---

### (e) \(\frac{x^{-2}}{y}\)

Simplify \(\frac{x^{-2}}{y}\). The negative exponent in \(x^{-2}\) moves \(x^2\) to the denominator:
\[
\frac{x^{-2}}{y} = \frac{1}{x^2 y}
\]

So, the simplified form is:
\[
\frac{1}{x^2 y}
\]

---

### (a) \((15x^7)(5x^{-2}) / x\)

First, simplify the numerator \((15x^7)(5x^{-2})\). Multiply the constants and use the exponent rule \(x^a \cdot x^b = x^{a+b}\):
\[
(15 \cdot 5)(x^7 \cdot x^{-2}) = 75x^{7 + (-2)} = 75x^5
\]

Now, divide by \(x\):
\[
\frac{75x^5}{x} = 75x^{5-1} = 75x^4
\]

So, the simplified form is:
\[
75x^4
\]

---

### (b) \(\frac{(15x^7)}{(5x^{-2})x}\)

First, simplify the fraction \(\frac{15x^7}{5x^{-2}}\). Divide the constants and subtract the exponents of \(x\):
\[
\frac{15}{5} \cdot \frac{x^7}{x^{-2}} = 3x^{7 - (-2)} = 3x^9
\]

Now, divide by \(x\):
\[
\frac{3x^9}{x} = 3x^{9-1} = 3x^8
\]

So, the simplified form is:
\[
3x^8
\]

---

Would you like further details or have any other questions? Here are five follow-up questions:

1. How do we handle negative exponents when simplifying fractions?
2. Can you explain the steps for multiplying expressions with exponents?
3. What is the rule for dividing terms with the same base in exponents?
4. How do you simplify expressions with both constants and variables?
5. How can exponent rules help in simplifying more complex algebraic expressions?

**Tip:** When working with exponents, always remember the rule that \(x^a \cdot x^b = x^{a+b}\) and \(x^a / x^b = x^{a-b}\).

Simplify the following expressions: (x/y)^(-2), x^(-2)/y, (15x^7)(5x^(-2)) / x, (15x^7) / (5x^(-2))x.

Learn how to simplify algebraic expressions involving negative exponents, multiplication, and division. This problem explores key exponent rules and simplifying techniques, suitable for students in Grades 9-10.

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