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Select the correct answer.

Evaluate the following expression when [tex] x = -4 [/tex] and [tex] y = 4 [/tex]:

[tex]\[ \frac{x^6 - x}{4y} \][/tex]

A. [tex] \frac{1,025}{4} \]

B. [tex] \frac{1,023}{4} \]

C. [tex] \frac{1,023}{4} \]

D. [tex] \frac{16,385}{4} \]


Sagot :

Sure, let's solve the given expression step-by-step and determine which option is correct.

We start with the expression:

[tex]\[ \frac{x^6 - x}{4y} \][/tex]

Given [tex]\( x = -4 \)[/tex] and [tex]\( y = 4 \)[/tex], we first evaluate [tex]\( x^6 - x \)[/tex]:

1. Calculate [tex]\( x^6 \)[/tex]:

[tex]\[ (-4)^6 = 4096 \][/tex]

2. Subtract [tex]\( x \)[/tex] from [tex]\( x^6 \)[/tex]:

[tex]\[ 4096 - (-4) = 4096 + 4 = 4100 \][/tex]

Next, we substitute [tex]\( y = 4 \)[/tex] into the denominator:

[tex]\[ 4y = 4 \cdot 4 = 16 \][/tex]

Now, we can write the reduced expression:

[tex]\[ \frac{4100}{16} \][/tex]

To simplify this fraction:

[tex]\[ \frac{4100}{16} = 256.25 \][/tex]

Now let's compare this result to the given options:

A. [tex]\(\frac{1025}{4} = 256.25\)[/tex]

B. [tex]\(\frac{1023}{4} = 255.75\)[/tex]

C. [tex]\(\frac{1023}{4} = 255.75\)[/tex]

D. [tex]\(\frac{16385}{4} = 4096.25\)[/tex]

From the options, option A, [tex]\(\frac{1025}{4}\)[/tex], equals 256.25, which matches our result.

Thus, the correct answer is:

[tex]\[ \boxed{\frac{1025}{4}} \][/tex]