2 to the power of x times 5 to the power of y = 80. What is x-y?

We are given the equation:

\[
2^x \times 5^y = 80
\]

To solve for \( x - y \), we can break down 80 into its prime factors:

\[
80 = 2^4 \times 5
\]

Now, rewrite the equation in terms of powers of 2 and 5:

\[
2^x \times 5^y = 2^4 \times 5^1
\]

This implies that the exponents of 2 and 5 on both sides must be equal:

- From the powers of 2: \( x = 4 \)
- From the powers of 5: \( y = 1 \)

Thus, we have:
\[
x = 4 \quad \text{and} \quad y = 1
\]

Now, compute \( x - y \):

\[
x - y = 4 - 1 = 3
\]

So, \( x - y = 3 \).

Would you like a more detailed explanation or have any questions?

Here are 5 related questions for practice:
1. Solve \( 3^x \times 2^y = 72 \) for \( x - y \).
2. If \( 2^{x+1} \times 3^y = 144 \), what is \( x - y \)?
3. Find the value of \( x \) and \( y \) if \( 6^x \times 5^y = 180 \).
4. Solve \( 4^x \times 25^y = 400 \) for \( x - y \).
5. If \( 3^x \times 5^{y-1} = 45 \), what is \( x + y \)?

**Tip:** When working with equations involving powers, try to express all terms using the same base to compare the exponents more easily.

Explore how to solve the equation 2^x * 5^y = 80 to find the difference x - y. This problem involves key algebraic concepts such as prime factorization, exponents, and the laws of exponents, making it ideal for students in Grades 8-10.

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