If `a gt 6,` which of the following is equivalent to `((2)/(a))/((1)/(a -2) + (1)/(a-6))` ?

Options

A. `2a ^(2) -16a +24`
B. `a (2a -8)`
C. `(a^(2) -8a +12)/(a ^(2) -4a)`
D. `(2a -80)/(a ^(2) - 8a +12)`

Answer

To simplify the expression \(\frac{2/a}{\frac{1}{a-2} + \frac{1}{a-6}}\), we can follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ \frac{2/a}{\frac{1}{a-2} + \frac{1}{a-6}} \]

### Step 2: Simplify the denominator
To simplify the denominator \(\frac{1}{a-2} + \frac{1}{a-6}\), we need to find a common denominator. The common denominator will be \((a-2)(a-6)\). Therefore, we rewrite the fractions:
\[ \frac{1}{a-2} = \frac{(a-6)}{(a-2)(a-6)} \quad \text{and} \quad \frac{1}{a-6} = \frac{(a-2)}{(a-2)(a-6)} \]
Now, we can add these two fractions:
\[ \frac{(a-6) + (a-2)}{(a-2)(a-6)} = \frac{2a - 8}{(a-2)(a-6)} \]

### Step 3: Substitute back into the expression
Now, we substitute this back into the original expression:
\[ \frac{2/a}{\frac{2a - 8}{(a-2)(a-6)}} \]

### Step 4: Rewrite as multiplication
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{2}{a} \times \frac{(a-2)(a-6)}{(2a - 8)} \]

### Step 5: Simplify the expression
Now, we can simplify this expression. The term \(2a - 8\) can be factored:
\[ 2a - 8 = 2(a - 4) \]
So we rewrite the expression:
\[ \frac{2(a-2)(a-6)}{a \cdot 2(a-4)} \]
Now, we can cancel the \(2\) in the numerator and denominator:
\[ \frac{(a-2)(a-6)}{a(a-4)} \]

### Final Result
Thus, the simplified expression is:
\[ \frac{(a-2)(a-6)}{a(a-4)} \]