Let's simplify the given expression:

$\left(\frac{y}{y^3}\right)^{-1}$

Step 1: Apply the negative exponent rule

The rule for negative exponents is:

$a^{-n} = \frac{1}{a^n}$

So, we apply this rule to the expression:

$\left(\frac{y}{y^3}\right)^{-1} = \frac{1}{\left(\frac{y}{y^3}\right)}$

Step 2: Simplify the denominator

Now, simplify the denominator:

$\frac{y}{y^3} = y^{1 - 3} = y^{-2}$

So, the expression becomes:

$\frac{1}{y^{-2}}$

Step 3: Apply the negative exponent again

Using the same exponent rule, $y^{-2}$ becomes:

$\frac{1}{y^{-2}} = y^2$

Final Answer:

$\left(\frac{y}{y^3}\right)^{-1} = y^2$

Let me know if you need more details or have any questions!

5 Related Questions:

1. What is the rule for simplifying expressions with negative exponents?
2. How do you simplify a fraction with exponents in both the numerator and denominator?
3. How do you handle an expression with multiple variables raised to powers?
4. What happens if the base is a negative number when raised to a negative exponent?
5. How can you use exponent rules to simplify complex algebraic expressions?

Tip:

When working with exponents, always remember to combine terms with the same base using addition or subtraction (depending on the operation).