Answer:
(a + 1/a)^3 = A^3 + 13.75.
Explanation:
To solve this problem, we first need to simplify the expression (a + 1/a)^3 using the identity (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
Let's start by expanding (a + 1/a)^3:(a + 1/a)^3 = a^3 + 3a^2(1/a) + 3a(1/a)^2 + (1/a)^3
We can simplify this expression using the fact that 1/a^2 = 1/a * 1/a:
(a + 1/a)^3 = a^3 + 3a + 3/a + 1/a^3
Now, we can substitute the given equation A + 1/a = 4:
(a + 1/a)^3 = A^3 + 3A + 3(1/A) + 1/a^3
We still need to find the value of a^3 + 1/a^3. To do this, we can use the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2), where a = a and b = 1/a:
a^3 + (1/a)^3 = (a + 1/a)(a^2 - a(1/a) + (1/a)^2)
a^3 + (1/a)^3 = (a + 1/a)(a^2 - 1 + 1/a^2)
But we know that A + 1/a = 4, so A^2 + 1/a^2 = (A + 1/a)^2 - 2 = 4^2 - 2 = 14. Substituting this in the previous expression gives:
a^3 + (1/a)^3 = (4)(14 - 1) = 52
Finally, substituting in the expression we derived earlier for (a + 1/a)^3 gives:
(a + 1/a)^3 = A^3 + 3A + 3(1/A) + 52
We know that A + 1/a = 4, so substituting this gives:
(a + 1/a)^3 = A^3 + 3(4) + 3(1/4) + 52 = A^3 + 13.75
Therefore, (a + 1/a)^3 = A^3 + 13.75.