If 2^x = 8^y, where x and y are positive integers, which of the following is equivalent to 8^(x+y)?

Rationale

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

To express 8^(x+y) in terms of base 2, we can rewrite 8 as 2^3. Thus, 8^(x+y) becomes (2^3)^(x+y), which simplifies to 2^(3(x+y)) or 2^(3x + 3y). Since 2^x = 8^y, we can substitute y in terms of x, leading us to 2^(3x).

A (2^(3x)) CORRECT

This choice is correct as demonstrated. By rewriting 8^(x+y) as (2^3)^(x+y), we simplify it to 2^(3x + 3y). Substituting y gives us an expression that ultimately equals 2^(3x).

B (2^(4x)) YOUR ANSWER

This option is incorrect because 2^(4x) does not derive from the simplification of 8^(x+y). The powers do not match; instead, 8^(x+y) equates to 2^(3x + 3y), which is not equal to 2^(4x) unless specific conditions on y are met.

C (2^(3x)^2) YOUR ANSWER

While this expression evaluates to 2^(6x), it does not accurately represent 8^(x+y). The calculation of 8^(x+y) leads to 2^(3x + 3y), not 2^(6x); thus, this option is not equivalent.

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

This choice is incorrect since it represents the sum of two different powers of 2, rather than a single exponential expression equivalent to 8^(x+y). The simplification of 8^(x+y) leads to a single power of 2, not a sum.

Conclusion

The expression 8^(x+y) simplifies to 2^(3x) when rewritten in base 2. Among the provided choices, only A) 2^(3x) matches this equivalence. The other options fail to accurately represent the relationship derived from the original equation involving 8 and its conversion to base 2. Understanding these transformations is crucial for solving exponential equations involving different bases.

If 2^x = 8^y, where x and y are positive integers, which of the following is equivalent to 8^(x+y)?

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

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