Question 7

8^(x+y) is equivalent to 2^(3x)

Given that \(2^x = 8^y\), we can express 8 in terms of base 2. Since \(8 = 2^3\), we rewrite \(8^y\) as \((2^3)^y = 2^{3y}\). This leads to the equation \(2^x = 2^{3y}\), allowing us to conclude that \(x = 3y\). To find \(8^{(x+y)}\), we can rewrite it as \(8^{(3y+y)} = 8^{4y}\), which can be expressed in base 2 as \((2^3)^{4y} = 2^{12y}\). Since \(x = 3y\), we can substitute \(y\) in terms of \(x\) to find that \(8^{(x+y)} = 2^{4x}\).

A) 2^(3x)

This option is incorrect because, while it seems related to the transformation of \(x\) in terms of \(y\), \(8^{(x+y)}\) actually evaluates to \(2^{4x}\) rather than \(2^{3x}\). The correct equivalence requires recognizing the relationship between \(x\) and \(y\) and the exponent calculation for \(8^{(x+y)}\).

B) 2^(4x)

This option is indeed correct. When we rewrite \(8^{(x+y)}\) as \((2^3)^{(x+y)}\) and simplify, we arrive at \(2^{3(x+y)}\). Substituting \(y\) in terms of \(x\) gives us \(2^{4x}\), confirming that this option accurately represents the equivalent expression for \(8^{(x+y)}\).

C) 2^(3x)^2

This option is incorrect because \(2^{(3x)^2}\) simplifies to \(2^{9x}\), which does not align with the evaluation of \(8^{(x+y)}\). The exponent structure does not match the requirements of the transformation from base 8 to base 2 in the context of \(x\) and \(y\).

D) 2^(3x)+ 2^x

This option is also incorrect. It presents a sum of two exponential expressions rather than a single equivalent exponent. The transformation of \(8^{(x+y)}\) does not result in a sum, and therefore this option does not reflect the proper calculation.

Conclusion

The correct answer is \(B) 2^{4x}\), which accurately reflects the transformation and relationship between \(x\) and \(y\) when calculating \(8^{(x+y)}\). All other options fail to capture the correct exponent structure, either misrepresenting the relationship between the variables or suggesting incorrect operations.