If (2^x)(2^y) = 8, what is the value of (x + y)?

Rationale

x + y = 3.

To solve the equation (2^x)(2^y) = 8, we can first express 8 as a power of 2, specifically 2^3. Using the property of exponents that states a^m * a^n = a^(m+n), we can rewrite the equation as 2^(x+y) = 2^3. This allows us to equate the exponents directly, leading to the conclusion that x + y = 3.

A (4) YOUR ANSWER

This choice suggests that the sum of x and y equals 4. However, substituting this value back into our equation would imply that 2^(x+y) = 2^4, which equals 16, not 8. Therefore, this option does not satisfy the original equation.

B (3) CORRECT

This is the correct answer, as shown in the explanation above. When we set 2^(x+y) equal to 2^3, we find that x + y must equal 3 to satisfy the equation (2^x)(2^y) = 8.

C (2) YOUR ANSWER

If x + y were to equal 2, substituting this back into the equation gives us 2^(x+y) = 2^2, which equals 4. This does not match the original equation where the right side equals 8, thus making this choice incorrect.

D (1) YOUR ANSWER

Choosing 1 for x + y would lead to 2^(x+y) = 2^1, which equals 2. This is much less than 8, indicating that this value does not satisfy the original equation and is therefore incorrect.

Conclusion

The sum of the exponents x and y in the equation (2^x)(2^y) = 8 must equal 3, as derived from equating the powers of 2. The only choice that satisfies this condition is option B, which confirms the relationship between the variables in the context of exponential equations. Other options fail to meet the criteria set by the original equation.

If (2^x)(2^y) = 8, what is the value of (x + y)?

x + y = 3.

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