Evaluate the following expression when [tex]x=-4[/tex] and [tex]y=4[/tex].
[tex]\frac{x^5-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 5 − x .
Calculate ( − 4 ) 5 = − 1024 .
Simplify the expression to 4 ( 4 ) − 1024 − ( − 4 ) = 16 − 1020 .
Simplify further to get the final answer: − 4 255 .
Explanation
Problem Setup We are given the expression 4 y x 5 − x and the values x = − 4 and y = 4 . Our goal is to evaluate the expression by substituting these values.
Substitution First, we substitute x = − 4 and y = 4 into the expression: 4 ( 4 ) ( − 4 ) 5 − ( − 4 )
Calculating (-4)^5 Next, we calculate ( − 4 ) 5 . Since ( − 4 ) 5 = − 4 × − 4 × − 4 × − 4 × − 4 = − 1024 , the expression becomes: 4 ( 4 ) − 1024 − ( − 4 )
Simplifying the Numerator Now, we simplify the numerator: − 1024 − ( − 4 ) = − 1024 + 4 = − 1020 . The expression is now: 4 ( 4 ) − 1020
Simplifying the Denominator Next, we simplify the denominator: 4 ( 4 ) = 16 . The expression is now: 16 − 1020
Final Calculation Finally, we divide − 1020 by 16 : 16 − 1020 = − 16 1020 = − 4 255 Since 255 × 4 = 1020 , we have 4 1020 = 255 . Thus, the simplified expression is − 4 255 . This is equivalent to − 16 1020 .
Checking Answer Choices The expression evaluates to − 16 1020 , which simplifies to − 4 255 . This is equal to − 16 1020 = − 63.75 . Comparing this to the answer choices, we see that − 4 1023 is the closest value. However, our calculation shows the answer should be − 16 1020 = − 4 255 .
We can express the answer choices as decimals to compare: A. 4 1023 = 255.75 B. 4 1025 = 256.25 C. − 4 1023 = − 255.75 D. 4 16385 = 4096.25 None of these match our result of − 16 1020 = − 63.75 . There must be an error in the provided answer choices or in the original problem statement. However, based on our calculations, the correct answer should be − 4 255 .
Let's re-evaluate the expression: 4 y x 5 − x = 4 ( 4 ) ( − 4 ) 5 − ( − 4 ) = 16 − 1024 + 4 = 16 − 1020 = − 4 255 .
Examples
Evaluating algebraic expressions is a fundamental skill in many areas of mathematics and science. For example, in physics, you might use an expression to calculate the force on an object based on its mass and acceleration. If the expression is F = ma , where F is force, m is mass, and a is acceleration, and you know that m = 5 kg and a = 2 m/s 2 , you can substitute these values into the expression to find the force: F = 5 × 2 = 10 N. Similarly, in economics, you might use an expression to calculate the profit of a business based on its revenue and costs. Substituting given values into algebraic expressions allows us to model and analyze real-world situations.
The evaluated expression 4 y x 5 − x for x = − 4 and y = 4 results in − 4 255 . This is the final outcome after completing the calculations. While the closest answer choice is C, − 4 1023 , our correct result is − 4 255 .
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