If 2^x is equal to 32, and 3^y is equal to 81, then what is xy?

To solve for \( xy \), we first need to determine the values of \( x \) and \( y \) individually.

### Step 1: Solve \( 2^x = 32 \)
We know that \( 32 = 2^5 \). So, by comparing the exponents:
\[
2^x = 2^5 \quad \Rightarrow \quad x = 5
\]

### Step 2: Solve \( 3^y = 81 \)
We know that \( 81 = 3^4 \). So, by comparing the exponents:
\[
3^y = 3^4 \quad \Rightarrow \quad y = 4
\]

### Step 3: Calculate \( xy \)
Now that we have \( x = 5 \) and \( y = 4 \), we can calculate \( xy \):
\[
xy = 5 \times 4 = 20
\]

Thus, the value of \( xy \) is \( \boxed{20} \).

Do you have any further questions or need more details?

Here are 5 related questions you might want to explore:
1. What is the solution to \( 2^{x+1} = 64 \)?
2. What is the value of \( 5^z = 625 \)?
3. What if \( 2^x = 64 \) and \( 5^y = 25 \)? What is \( xy \)?
4. How do you solve logarithmic equations like \( \log_2(32) = x \)?
5. What is the relationship between exponential and logarithmic functions?

**Tip**: To solve exponential equations like \( a^x = b \), try expressing \( b \) as a power of \( a \) if possible, and then compare exponents.

Learn how to solve for xy given that 2^x equals 32 and 3^y equals 81. This step-by-step solution explains the properties of exponents and demonstrates how to find values for x and y. Suitable for students in Grades 6-8.

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