Final answer:
To express the given expression as a single fraction, find a common denominator among the terms. Rewrite each term with the common denominator. Combine the terms and simplify the numerator. The expression can be expressed as (a³ - 5a² + 3a + 18)/(a - 6)(a + 6)(36 - a²).
Step-by-step explanation:
To express the given expression as a single fraction, we need to find a common denominator. The denominators in the expression are (a - 6), (a + 6), and (36 - a²). The common denominator for these terms would be (a - 6)(a + 6)(36 - a²). Now, we can rewrite each term with the common denominator:
a/(a - 6) = a(a + 6)(36 - a²)/(a - 6)(a + 6)(36 - a²)
3/(a + 6) = 3(a - 6)(36 - a²)/(a - 6)(a + 6)(36 - a²)
a²/(36 - a²) = a²(a - 6)(a + 6)/(a - 6)(a + 6)(36 - a²)
Now we can combine these terms:
a(a + 6)(36 - a²)/(a - 6)(a + 6)(36 - a²) - 3(a - 6)(36 - a²)/(a - 6)(a + 6)(36 - a²) + a²(a - 6)(a + 6)/(a - 6)(a + 6)(36 - a²)
Canceling out the common factors in the numerator and denominator, we get:
(a(a + 6) - 3(a - 6) + a²(a - 6))/(a - 6)(a + 6)(36 - a²)
Expanding and simplifying the numerator, we get:
(a² + 6a - 3a + 18 + a³ - 6a²)/(a - 6)(a + 6)(36 - a²)
Combining like terms:
(a³ - 5a² + 3a + 18)/(a - 6)(a + 6)(36 - a²)
Therefore, the expression can be expressed as a single fraction: (a³ - 5a² + 3a + 18)/(a - 6)(a + 6)(36 - a²)