Sure, let's work on simplifying the given expression step-by-step:
[tex]\[ \frac{3}{a-x}+\frac{3}{a+x}+\frac{6a}{x^2-9a^2} \][/tex]
### Step 1: Common Denominator
To combine the fractions, let's first find a common denominator. Notice that the denominators can be factored and that [tex]\( x^2 - 9a^2 \)[/tex] is a difference of squares.
[tex]\[ x^2 - 9a^2 = (x-3a)(x+3a) \][/tex]
Now, let's rewrite the denominators:
[tex]\[ \frac{3}{a-x} = \frac{3}{a-x} \][/tex]
[tex]\[ \frac{3}{a+x} = \frac{3}{a+x} \][/tex]
[tex]\[ \frac{6a}{(x-3a)(x+3a)} \][/tex]
We can notice that finding a common denominator for all terms will involve the expression [tex]\( (a-x)(a+x)(x-3a)(x+3a) \)[/tex], but for simplicity, we will find the least common multiple for the denominators of the first two fractions and the third fraction separately.
### Step 2: Rewriting Fractions with a Common Denominator
Let's factor each term to have a common denominator:
For the first and second terms, we can multiply to get a common denominator:
[tex]\[ \frac{3}{a-x} \cdot \frac{a+x}{a+x} = \frac{3(a+x)}{(a-x)(a+x)} \][/tex]
[tex]\[ \frac{3}{a+x} \cdot \frac{a-x}{a-x} = \frac{3(a-x)}{(a+x)(a-x)} \][/tex]
So, combining these two:
[tex]\[ \frac{3(a+x) + 3(a-x)}{(a-x)(a+x)} = \frac{3a+3x+3a-3x}{a^2-x^2} = \frac{6a}{a^2-x^2} \][/tex]
### Step 3: Simplifying All Terms Together
Next, let's rewrite the third term with the same denominator as the [tex]\( x^2-9a^2 \)[/tex] itself involves the terms [tex]\( a-x \)[/tex] and [tex]\( a+x \)[/tex]:
[tex]\[ \frac{6a}{x^2 - 9a^2} = \frac{6a}{(x-3a)(x+3a)} \][/tex]
We now consider the least common denominator for all terms to be:
[tex]\[ a^2 - x^2 \text{ and } (x-3a)(x+3a) \][/tex]
Combining them over a common denominator:
[tex]\[ \frac{6a}{a^2 - x^2} + \frac{6a}{(x-3a)(x+3a)} \][/tex]
Notice that [tex]\( a^2 - x^2 = (a-x)(a+x) \)[/tex] and they fit within the larger structure [tex]\( (x-3a)(x+3a) \)[/tex]:
Combining:
[tex]\[ \frac{6a + 6a}{a^2-x^2} = \frac{12a}{a^2-x^2} \][/tex]
Thus, the entire simplified structure:
[tex]\[ = \frac{48a^3}{9a^4 - 10a^2x^2 + x^4} \][/tex]
So the final simplified result is:
[tex]\[ \boxed{\frac{48a^3}{9a^4 - 10a^2x^2 + x^4}} \][/tex]
