https://file.aiquickdraw.com/mathbot/file/1727279409731ul7987bh.jpg

We are given the equation:

\[
\frac{2^x}{2^y} = \frac{1}{8}
\]

and

\[
x + y = -3
\]

We are asked to find the value of \( x - y \).

### Step 1: Simplify the first equation
The left-hand side of the equation can be simplified as:

\[
\frac{2^x}{2^y} = 2^{x - y}
\]

Thus, the equation becomes:

\[
2^{x - y} = \frac{1}{8}
\]

Now, express \( \frac{1}{8} \) as a power of 2. Since \( 8 = 2^3 \), we have:

\[
\frac{1}{8} = 2^{-3}
\]

Thus, the equation simplifies to:

\[
2^{x - y} = 2^{-3}
\]

### Step 2: Solve for \( x - y \)
Since the bases are the same, we can equate the exponents:

\[
x - y = -3
\]

### Step 3: Conclusion
The value of \( x - y \) is \( \boxed{-3} \).

If you need more details or have any questions, feel free to ask!

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Here are 5 related questions to expand your understanding:
1. How do we simplify fractions involving powers of the same base?
2. What are the properties of exponents that allowed us to equate \( 2^{x - y} \) and \( 2^{-3} \)?
3. Can you solve a similar problem where \( \frac{3^x}{3^y} = \frac{1}{9} \) and \( x + y = -4 \)?
4. What happens if \( x + y \) is a positive number in this type of problem?
5. How can we use logarithms to solve problems involving exponential equations?

**Tip:** When solving problems with exponents, remember to express numbers as powers of the same base whenever possible to simplify the equation.

If 2^x / 2^y = 1/8 and x + y = -3, what is the value of x - y?

Learn how to solve the equation 2^x / 2^y = 1/8 given that x + y = -3, using properties of exponents. This problem involves algebraic manipulation and is suitable for students in Grades 9-11.

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