Select the correct answer.
Evaluate the following expression when [tex]x=-4[/tex] and [tex]y=4[/tex].
[tex]\frac{x^6-x}{4 y}[/tex]
A. [tex]\frac{1,023}{4}[/tex]
B. [tex]\frac{1,025}{4}[/tex]
C. [tex]-\frac{1,023}{4}[/tex]
D. [tex]\frac{16,385}{4}[/tex]
Answer (2)
Substitute x = − 4 and y = 4 into the expression 4 y x 6 − x .
Calculate ( − 4 ) 6 = 4096 .
Simplify the expression to 4 ( 4 ) 4096 − ( − 4 ) = 16 4100 .
Divide to get the final answer: 16 4100 = 4 1025 .
Explanation
Understanding the problem We are given the expression 4 y x 6 − x and the values x = − 4 and y = 4 . Our goal is to evaluate the expression by substituting the given values.
Substituting the values First, we substitute x = − 4 and y = 4 into the expression: 4 ( 4 ) ( − 4 ) 6 − ( − 4 )
Calculating (-4)^6 Next, we calculate ( − 4 ) 6 . Since ( − 4 ) 6 = ( − 4 ) × ( − 4 ) × ( − 4 ) × ( − 4 ) × ( − 4 ) × ( − 4 ) = 4096 . So we have: 4 ( 4 ) 4096 − ( − 4 )
Simplifying numerator and denominator Now, we simplify the numerator: 4096 − ( − 4 ) = 4096 + 4 = 4100 . And we simplify the denominator: 4 ( 4 ) = 16 . So the expression becomes: 16 4100
Dividing to get the final answer Finally, we divide 4100 by 16: 16 4100 = 256.25 We can also write this as a fraction: 16 4100 = 4 1025
Examples
Evaluating algebraic expressions is a fundamental skill in mathematics and has practical applications in various fields. For instance, in physics, you might use such expressions to calculate the force acting on an object given its mass and acceleration. In finance, you could evaluate expressions to determine the return on an investment based on different interest rates and time periods. Understanding how to substitute values into expressions and simplify them allows you to model and solve real-world problems effectively.
After substituting x = − 4 and y = 4 into the expression 4 y x 6 − x , we found the result to be 4 1025 . The correct option from the choices given is B: 4 1 , 025 .
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