If `((a)/(b))+((b)/(a))=2`, then find `((a)/(b))^(100)-((b)/(a))^(10)`

Options

A. `(2^(20)-1)/(2^(10))`
B. 2
C. 0
D. `(2^(20)+1)/(2^(10))`

Answer

To solve the equation \(\frac{a}{b} + \frac{b}{a} = 2\) and find the value of \(\left(\frac{a}{b}\right)^{100} - \left(\frac{b}{a}\right)^{10}\), we can follow these steps:

### Step 1: Let \(\frac{a}{b} = T\)
We start by substituting \(\frac{a}{b}\) with \(T\). Consequently, \(\frac{b}{a} = \frac{1}{T}\).

### Step 2: Rewrite the equation
Now, we can rewrite the equation:
\[ T + \frac{1}{T} = 2 \]

### Step 3: Multiply through by \(T\)
To eliminate the fraction, we multiply the entire equation by \(T\):
\[ T^2 + 1 = 2T \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ T^2 - 2T + 1 = 0 \]

### Step 5: Factor the quadratic
This can be factored as:
\[ (T - 1)^2 = 0 \]

### Step 6: Solve for \(T\)
Setting the factor equal to zero gives:
\[ T - 1 = 0 \implies T = 1 \]

### Step 7: Substitute back to find \(\frac{a}{b}\) and \(\frac{b}{a}\)
Since \(T = \frac{a}{b}\), we have:
\[ \frac{a}{b} = 1 \quad \text{and} \quad \frac{b}{a} = 1 \]

### Step 8: Calculate the required expression
Now we need to find:
\[ \left(\frac{a}{b}\right)^{100} - \left(\frac{b}{a}\right)^{10} \]
Substituting the values we found:
\[ 1^{100} - 1^{10} = 1 - 1 = 0 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{0} \]