The rule states that to make a negative exponent positive when dealing with a fraction, you simply flip the fraction upside down.

Think of it like this: The negative sign on the exponent is like a switch. Flip the fraction (the base), and the switch turns off (becomes positive).

Why Does It Work?

You don’t have to memorize this blindly; the logic comes from the standard Negative Exponent Rule we discussed earlier.

Let’s look at $(\frac{x}{y})^{-2}$.

1. Distribute the negative: According to the rules of exponents, this equals $\frac{x^{-2}}{y^{-2}}$.

2. Move the terms:

   • The $x^{-2}$ on top wants to go downstairs to become positive.

   • The $y^{-2}$ on the bottom wants to go upstairs to become positive.

3. Result: You end up with $\frac{y^2}{x^2}$, which is the same as $(\frac{y}{x})^2$.

The Reciprocal Rule just lets you skip the middle steps!

How to Use It: 3 Simple Steps

When you see a fraction inside parentheses with a negative exponent outside:

1. FLIP: Turn the fraction into its reciprocal (numerator becomes denominator, denominator becomes numerator).

2. SWITCH: Change the exponent from negative to positive.

3. SOLVE: Distribute the exponent to both the top and bottom numbers.

Example 1: Numbers

Problem: Simplify $(\frac{2}{3})^{-3}$

1. Flip & Switch: Change $\frac{2}{3}$ to $\frac{3}{2}$ and change $-3$ to $+3$. 
   $$(\frac{3}{2})^3$$
2. Solve: Apply the power of 3 to both numbers. 
   $$\frac{3^3}{2^3} = \frac{27}{8}$$

Example 2: Variables

Problem: Simplify $(\frac{4x}{y})^{-2}$

1. Flip & Switch: 
   $$(\frac{y}{4x})^2$$
2. Solve: Remember to square everything inside, including the coefficient 4. 
   $$\frac{y^2}{4^2 x^2} = \frac{y^2}{16x^2}$$

Pro-Tip: The “One-Step” Flip

Sometimes you will see a problem that looks like a chain of variables, not a clean fraction in parentheses. You can still use the reciprocal concept individually.

Problem: $\frac{a^{-3} b^2}{c^{-5}}$

• $a^{-3}$: Negative. Needs to move down.

• $b^2$: Positive. stays where it is!

• $c^{-5}$: Negative. Needs to move up.

Result: $\frac{b^2 c^5}{a^3}$

Key Takeaway: Only move the specific pieces that have a “ticket” (a negative exponent). If a variable has a positive exponent, it is happy where it is—leave it alone!