To simplify the given expression:

$\frac{y^8 y^{-3}}{(y^2)^3}$

Step-by-Step Solution:

1. Combine the powers in the numerator: Using the product rule of exponents $a^m \cdot a^n = a^{m+n}$: $y^8 y^{-3} = y^{8 + (-3)} = y^5$

   The numerator simplifies to: $y^5$

2. Simplify the denominator: Using the power rule of exponents $(a^m)^n = a^{m \cdot n}$: $(y^2)^3 = y^{2 \cdot 3} = y^6$

   The denominator becomes: $y^6$

3. Simplify the entire expression: Using the quotient rule of exponents $\frac{a^m}{a^n} = a^{m-n}$: $\frac{y^5}{y^6} = y^{5 - 6} = y^{-1}$

4. Express the final answer: A negative exponent means reciprocal: $y^{-1} = \frac{1}{y}$

Final Answer:

$\frac{1}{y}$

---

Would you like further clarification or additional examples? 😊

5 Related Questions:

1. How do you simplify expressions with variables and negative exponents?
2. What are the key exponent rules and how are they applied in different scenarios?
3. Can you explain how to simplify $(x^3 y^{-2})^2$?
4. What happens when dividing expressions with the same base and different exponents?
5. How do fractional exponents differ from negative exponents in simplifications?

Tip:

Always apply exponent rules step by step, and remember that a negative exponent represents the reciprocal of the base raised to the positive exponent!