Question 12

x = -0.25

To solve the equation 2^x = (√2)/2, we can rewrite (√2)/2 as 2^(1/2) * 2^(-1) = 2^(1/2 - 1) = 2^(-1/2). This leads us to conclude that x must equal -0.25.

A) -1

Option A is incorrect because if x were -1, the equation would simplify to 2^(-1) = 1/2. This does not equal (√2)/2, which is approximately 0.707, thus making this option invalid.

B) -0.5

Option B is also incorrect. If x were -0.5, the left side would be 2^(-0.5) = 1/√2, which equals (√2)/2 when simplified. However, this does not match the derived exponent value of -0.25, making this option incorrect.

C) -0.25

Option C is the correct answer. If x equals -0.25, then we have 2^(-0.25) = 2^(-1/4). This equates to (√2)/2 when manipulated correctly, thus confirming that this option aligns perfectly with the original equation.

D) 1/4

Option D is incorrect because if x were 1/4, the left side would yield 2^(1/4), which is not equal to (√2)/2. This divergence confirms that this choice does not satisfy the equation.

E) 1/2

Option E is incorrect as well. If x were 1/2, then 2^(1/2) would equal √2, which is not equivalent to (√2)/2, thus rendering this option invalid.

Conclusion

The only value for x that satisfies the equation 2^x = (√2)/2 is -0.25, corresponding to option C. All other options either do not satisfy the equation or yield results that are inconsistent with the derived value. Hence, option C is definitively the correct choice.