To simplify the expression [tex]\(\left(\frac{1}{2}\right)^3 - 6 + \sqrt{64}\)[/tex], we will follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
1. Compute the exponent:
[tex]\[
\left(\frac{1}{2}\right)^3 = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{8}
\][/tex]
2. Compute the square root:
[tex]\[
\sqrt{64} = 8
\][/tex]
3. Combine all parts together:
[tex]\[
\left(\frac{1}{2}\right)^3 - 6 + \sqrt{64} = \frac{1}{8} - 6 + 8
\][/tex]
4. Perform the addition and subtraction from left to right:
[tex]\[
\frac{1}{8} - 6 + 8 = \frac{1}{8} - \frac{48}{8} + \frac{64}{8}
\][/tex]
[tex]\[
= \frac{1 - 48 + 64}{8}
\][/tex]
[tex]\[
= \frac{17}{8}
\][/tex]
\]
= 2.125
\]
The simplified result of the given expression is [tex]\(2.125\)[/tex], which does not directly correspond to a fraction option in the multiple-choice answers. Therefore, the final answer is neither [tex]\(-\frac{5}{8}\)[/tex], [tex]\(-\frac{47}{64}\)[/tex], [tex]\(\frac{5}{8}\)[/tex], nor [tex]\(\frac{47}{64}\)[/tex]. Given the numerical value, no exact match is found in the provided options. The correct answer derived is:
[tex]\[
2.125
\][/tex]
