Mathematical Expressions Evaluation and Simplification
The image contains two different mathematical expressions that need to be evaluated or simplified:
1. Evaluate \(\cos^{-1}(\frac{1}{2}+\frac{\sqrt{3}}{2})\)
2. Simplify \(\frac{4^{-2} \times 2x - 2}{4x - 3 \times 0.5}\)
Starting with the first one:
1. Evaluate \(\cos^{-1}(\frac{1}{2}+\frac{\sqrt{3}}{2})\)
The expression inside the cosine inverse function is not possible since the cosine values are constrained between -1 and 1, and \(\frac{1}{2} + \frac{\sqrt{3}}{2}\) exceeds this range. Hence, the expression given is incorrect as it stands. If this is a mistake and you meant \(\cos^{-1}(\frac{1}{2})\) or \(\cos^{-1}(\frac{\sqrt{3}}{2})\), the results would be \(60^{\circ}\) or \(30^{\circ}\) respectively because cos(60°) = 1/2 and cos(30°) = √3/2.
2. Simplify \(\frac{4^{-2} \times 2x - 2}{4x - 3 \times 0.5}\)
First simplify \(4^{-2}\). This is the same as \(1/4^{2}\) which equals \(1/16\).
The expression now is:
\(\frac{1/16 \times 2x - 2}{4x - 1.5}\)
This simplifies to:
\(\frac{(1/16) \times 2x - 2}{4x - 1.5}\)
\(\frac{(1/8)x - 2}{4x - 1.5}\)
There aren't any common factors between the numerator and the denominator, so this expression is simplified as much as possible given the current form. If additional context is provided or if there are any restrictions on the values of \(x\), further simplification may be possible.
