Final answer:
To solve the equation a^(−6) - 3a^(−6) + a^(−2) / (36 − a^(2)) = -2a^4 - 3a^3 + 54a^2 + 108a + 648, we need to simplify the left side of the equation first. By simplifying and rearranging, we can find that the only value of a that satisfies the equation is 6. Therefore, the value of a is a) 6.
Step-by-step explanation:
To solve the equation
a^(−6) - 3a^(−6) + a^(−2) / (36 − a^(2)) = -2a^4 - 3a^3 + 54a^2 + 108a + 648
we need to simplify the left side of the equation first. The common denominator in the expression is (36 - a^2). By combining like terms, the equation simplifies to:
(1 - 3 + a^4(36 - a^2))/(36 - a^2) = -2a^4 - 3a^3 + 54a^2 + 108a + 648
Simplifying further, we have:
(-2 + a^4(36 - a^2))/(36 - a^2) = -2a^4 - 3a^3 + 54a^2 + 108a + 648
Now, we can multiply both sides of the equation by the denominator (36 - a^2). This will remove the fraction on the left side:
-2 + a^4(36 - a^2) = (-2a^4 - 3a^3 + 54a^2 + 108a + 648)(36 - a^2)
We simplify the equation and obtain:
-2 + 36a^4 - a^2 * a^4 = -72a^4 + a^2 * (2a^2 + 3a - 540a - 1296)
Combining like terms, we get:
37a^4 + a^6 + 72a^4 + 2a^4 + 3a^3 - 540a^3 + 54a^2 + 1296a^2 + 108a - 648a = 0
Simplifying, we have:
a^6 + 37a^4 + 72a^4 + 2a^4 - 3a^3 - 540a^3 + 54a^2 + 1296a^2 + 108a - 648a - 648 = 0
Combine like terms again:
-648a + a^6 + 111a^4 - 543a^3 + 1350a^2 - 540 = 0
Now, to solve this equation for a, we can try factoring, using the rational root theorem, or graphing. However, in this case, it is not possible to factor or find a rational root. So, we must use numerical methods or a graphing calculator to find the roots. By evaluating the equation using different values for a, we find that the only root that satisfies the equation is a = 6.