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Express 2x4(2x²)−4−4(x³)−2 in the form ax18 + bx6, where a, b ∈ ℤ. Find the value of a.

a) 18
b) 16
c) 14
d) 12

1 Answer

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Final answer:

By applying negative exponent rules, we simplify the given expression, which results in terms not aligned with ax^18 + bx^6. Thus, the coefficients a and b are effectively 0, indicating the expression cannot be represented in the desired form with integer coefficients.

Step-by-step explanation:

To simplify the expression 2x^4(2x^2)-4 - 4(x^3)-2, we have to apply the exponent rules appropriately.

First, we simplify each term by applying the negative exponent rule, which states that a-n = 1/an.

2x^4(2x^2)-4 = 2x4 × (1/24x^8)

Then, we subtract the second term, 4(x^3)-2 = 4 × (1/x^6).

By combining like terms and simplifying, we end up with:

2x^4 × (1/24x^8) - 4 × (1/x^6) = 2x^4-8 × 1/16 - 4/x^6 = 1/8x^(-4) - 4x^(-6)

Finally, we express the result in the form ax^18 + bx^6, knowing that the exponents must add up to 18 and 6 respectively. The only way this is possible is if the terms we have are already part of the final expression, as x^(-4) would correspond to a term involving x18 in the denominator, and x^(-6) a term involving x6 in the denominator. This means that a and b are effectively 0, as the expression cannot be put into the desired form ax18 + bx6 with integer coefficients.

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