- Substitute \(x = -4\) and \(y = 4\) into the expression \(\frac{x^6 - x}{4y}\).
- Calculate the numerator: \(x^6 - x = (-4)^6 - (-4) = 4096 + 4 = 4100\).
- Calculate the denominator: \(4y = 4 \times 4 = 16\).
- Divide the results to obtain the final value: \(\frac{4100}{16} = \frac{1025}{4}\).
- The final answer is \(\boxed{\frac{1025}{4}}\).
Explanation
- Evaluating the Expression
The objective is to evaluate the expression \(\frac{x^6 - x}{4y}\) given \(x = -4\) and \(y = 4\). (Note: Based on the provided options, it appears the expression intended for evaluation is \(\frac{x^6 - x}{4y}\), as this leads to one of the provided solutions).
First, we substitute \(x = -4\) into the numerator:
\[x^6 - x = (-4)^6 - (-4)\]
Calculate the power of -4:
\[(-4)^6 = 4096\]
Substitute this value back into the expression for the numerator:
\[4096 - (-4) = 4096 + 4 = 4100\]
Next, substitute \(y = 4\) into the denominator:
\[4y = 4 \times 4 = 16\]
Now, compute the full fraction:
\[\frac{4100}{16}\]
Simplifying the fraction by dividing both the numerator and the denominator by 4:
\[\frac{4100 \div 4}{16 \div 4} = \frac{1025}{4}\]
Comparing this result with the given options, we confirm that \(\frac{1025}{4}\) is one of the choices.
Examples
In many scientific and engineering fields, evaluating complex formulas with specific variables is a daily task. For instance, in structural engineering, stress analysis formulas involve variables that represent physical dimensions or forces. Calculating these values accurately ensures the safety of structures like bridges or buildings. Mastering algebraic substitution is a foundational skill for any professional working with quantitative models and technical designs.