Black Chandler
07/28/2024 · Senior High School
\[ 3^{x}-3^{y}=72 \] where ax and \( y \) are posibuve integer. Find \( x \) and \( y \)
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Tutor-Verified Answer
Step-by-step Solution
To solve the equation \( 3^x - 3^y = 72 \) where \( x \) and \( y \) are positive integers, we can factor the left-hand side:
\[
3^y(3^{x-y} - 1) = 72
\]
This implies that \( 3^y \) must be a divisor of 72. We can find the possible values of \( 3^y \) by determining the powers of 3 that divide 72.
The prime factorization of 72 is:
\[
72 = 2^3 \times 3^2
\]
The possible values for \( 3^y \) are \( 1 \) (when \( y = 0 \)), \( 3 \) (when \( y = 1 \)), and \( 9 \) (when \( y = 2 \)). However, since \( y \) must be a positive integer, we only consider \( 3 \) and \( 9 \).
### Case 1: \( 3^y = 3 \) (i.e., \( y = 1 \))
Substituting \( y = 1 \) into the equation:
\[
3^1(3^{x-1} - 1) = 72
\]
This simplifies to:
\[
3(3^{x-1} - 1) = 72
\]
Dividing both sides by 3:
\[
3^{x-1} - 1 = 24
\]
Adding 1 to both sides:
\[
3^{x-1} = 25
\]
Since \( 25 \) is not a power of 3, there are no integer solutions in this case.
### Case 2: \( 3^y = 9 \) (i.e., \( y = 2 \))
Substituting \( y = 2 \) into the equation:
\[
3^2(3^{x-2} - 1) = 72
\]
This simplifies to:
\[
9(3^{x-2} - 1) = 72
\]
Dividing both sides by 9:
\[
3^{x-2} - 1 = 8
\]
Adding 1 to both sides:
\[
3^{x-2} = 9
\]
Since \( 9 = 3^2 \), we have:
\[
x - 2 = 2
\]
Thus:
\[
x = 4
\]
### Conclusion
The values of \( x \) and \( y \) are:
\[
x = 4, \quad y = 2
\]
Therefore, the solution is:
\[
\boxed{4}
\] and \( y = 2 \).
Quick Answer
The solution is \( x = 4 \) and \( y = 2 \).
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