If 2 ^ x = (√(2))/2 what is the value of x ?

Rationale

x = -0.25.

To solve the equation (2^x = frac{sqrt{2}}{2}), we can rewrite (frac{sqrt{2}}{2}) as (2^{1/2 - 1}) or (2^{-0.5}). Setting the exponents equal gives us (x = -0.25).

A (-1) YOUR ANSWER

If (x = -1), then (2^x = 2^{-1} = frac{1}{2}), which does not equal (frac{sqrt{2}}{2}). Therefore, this choice is incorrect.

B (-0.5) YOUR ANSWER

If (x = -0.5), then (2^x = 2^{-0.5} = frac{1}{sqrt{2}}), which is mathematically equivalent to (frac{sqrt{2}}{2}). However, we are looking for a value that equates (2^x) directly to (frac{sqrt{2}}{2}), which requires (x) to be (-0.25).

C (-0.25) CORRECT

This value is correct since substituting (x = -0.25) yields (2^{-0.25} = 2^{(1/2 - 1)} = frac{sqrt{2}}{2}), matching the right side of the original equation.

D (1/4) YOUR ANSWER

If (x = frac{1}{4}), then (2^x = 2^{0.25}), which equals (sqrt[4]{2}). This does not equal (frac{sqrt{2}}{2}), making this choice incorrect.

E (1/2) YOUR ANSWER

If (x = frac{1}{2}), then (2^x = 2^{0.5} = sqrt{2}). This value also does not equal (frac{sqrt{2}}{2}), so this choice is incorrect.

Conclusion

The equation (2^x = frac{sqrt{2}}{2}) simplifies to (2^x = 2^{-0.5}), leading to (x = -0.25) as the solution. Other choices do not satisfy the equation, demonstrating the unique solution provided by (-0.25).

If 2 ^ x = (√(2))/2 what is the value of x ?

x = -0.25.

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