A/(Bn) = r^(1-n). I am asked to solve for r. I started by multiplying exponents on both sides with -1 but my answer is wrong. what did i do wrong here? explain the process.

It looks like you started by multiplying exponents on both sides by \(-1\), but that's not the correct first step for solving the equation for \(r\). Let's carefully go through the correct process.

### Given Equation:
\[
\frac{A}{B^n} = r^{1-n}
\]

You are solving for \(r\), so let's break this down step by step:

### Step 1: Isolate \(r^{1-n}\)

Start by multiplying both sides of the equation by \(B^n\) to eliminate the denominator:
\[
A = B^n \cdot r^{1-n}
\]

### Step 2: Isolate \(r\)

Now, you want to get rid of the exponent on \(r\), so you need to isolate \(r\) first by dividing both sides by \(B^n\):
\[
\frac{A}{B^n} = r^{1-n}
\]
But since this is exactly what we started with, let's simplify the solution by focusing on the exponent.

### Step 3: Raise both sides to the power of \(\frac{1}{1-n}\)

To isolate \(r\), you need to eliminate the exponent \((1-n)\). The correct operation is to raise both sides of the equation to the reciprocal of that exponent:
\[
\left( \frac{A}{B^n} \right)^{\frac{1}{1-n}} = r
\]

This operation removes the exponent from \(r\), leaving us with:
\[
r = \left( \frac{A}{B^n} \right)^{\frac{1}{1-n}}
\]

### Conclusion:

The mistake in your original approach was multiplying exponents by \(-1\), which doesn't correctly solve the equation. Instead, you need to isolate the variable with the exponent and use the reciprocal of the exponent to solve for \(r\).

### Final Answer:
\[
r = \left( \frac{A}{B^n} \right)^{\frac{1}{1-n}}
\]

---

Let me know if you would like further clarification or details!

Here are 5 questions to expand on this:

1. What is the reciprocal of an exponent and why is it used to isolate variables?
2. How do you handle equations with exponents that contain variables?
3. What steps can you take to check your solution after solving for a variable?
4. Can you express \(r\) in a simplified form if \(n = 2\)?
5. How would you solve for \(B\) if that were the unknown variable?

**Tip:** Always work to isolate the variable you are solving for before manipulating exponents!

A/(Bn) = r^(1-n). I am asked to solve for r. I started by multiplying exponents on both sides with -1 but my answer is wrong. What did I do wrong here? Explain the process.

Learn how to solve the exponential equation A/(Bn) = r^(1-n) with a step-by-step guide. This solution covers key algebraic concepts such as exponent manipulation and solving for variables. Ideal for students in Grades 9-12 looking to deepen their understanding of exponential equations.

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