\( \left( \begin{array} { l } { 1 } \\ { 2 } \end{array} \right) ^ { x - 5 } = 64 \)

To solve the equation \( \left( \begin{array} { l } { 1 } \\ { 2 } \end{array} \right) ^ { x - 5 } = 64 \), we can recognize that the structure is depicting powers. In this case, the left side suggests a vector or a base being raised to a power, but for simplification, let's treat it as numeric. Notice that \( 64 \) is equivalent to \( 2^6 \). To find \( x \), set \( x - 5 = 6 \) (since \( 64 \) needs to be \( 2 \) raised to some power). Solving this gives \( x = 11 \). Thus, you will find \( x = 11 \) as the solution to your equation! Now, let's dive into some insights! Did you know that powers of two are fundamental in computer science, particularly in programming and digital electronics? They form the basis for binary systems, which are essential for everything from coding to data processing. Learning about powers of two can enhance your computational skills and make tackling problems much easier! For further exploration, consider delving into the relationships between exponents and logarithms. Understanding this can deepen your insight into how numbers interact and give you tools to solve various equations with ease. You could start with introductory algebra texts or even online courses that focused on logarithmic functions; they often reveal interesting applications!