Question 36

3

To solve for the value of \(x + y\) in the equation \((2^x)(2^y) = 8\), we first rewrite 8 as a power of 2. Since \(8 = 2^3\), we can equate the exponents: \(x + y = 3\).

A) 4

This option suggests that the sum of \(x\) and \(y\) equals 4. Given that \(2^x \cdot 2^y = 2^{x+y}\) must equal \(2^3\), \(x + y\) cannot equal 4, making this option incorrect.

B) 3

This option correctly identifies that \(x + y = 3\). By rewriting the original equation as \(2^{x+y} = 2^3\), we see that the equality holds true, confirming that this option is indeed correct.

C) 2

A value of 2 for \(x + y\) would imply that \(2^{x+y} = 2^2\), which does not satisfy the original equation \((2^x)(2^y) = 8\). Therefore, this option is incorrect.

D) 1

This option suggests that \(x + y = 1\). Similar to option C, if \(x + y\) were to equal 1, we would have \(2^{x+y} = 2^1\), which is not equal to 8. Hence, this option is incorrect as well.

Conclusion

The correct answer is 3, as it accurately reflects the sum of the exponents that result in the equation being true. All other options fail to meet this condition, thus confirming that \(x + y\) must equal 3 for \((2^x)(2^y) = 8\) to hold.