If `((a)/(b))^((5)/(14))+ ((b)/(a))^((5)/(14))=6` then find the value of `((a)/(b))^((5)/(7))+((b)/(a))^((5)/(7))`

Answer

To solve the equation \(\left(\frac{a}{b}\right)^{\frac{5}{14}} + \left(\frac{b}{a}\right)^{\frac{5}{14}} = 6\) and find the value of \(\left(\frac{a}{b}\right)^{\frac{5}{7}} + \left(\frac{b}{a}\right)^{\frac{5}{7}}\), we can follow these steps:

### Step 1: Let \(x = \left(\frac{a}{b}\right)^{\frac{5}{14}}\)
We can rewrite the given equation in terms of \(x\):
\[ x + \frac{1}{x} = 6 \]

### Step 2: Multiply both sides by \(x\)
This gives us:
\[ x^2 + 1 = 6x \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ x^2 - 6x + 1 = 0 \]

### Step 4: Use the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \(a = 1\), \(b = -6\), and \(c = 1\):
\[ x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{6 \pm \sqrt{36 - 4}}{2} \]
\[ x = \frac{6 \pm \sqrt{32}}{2} \]
\[ x = \frac{6 \pm 4\sqrt{2}}{2} \]
\[ x = 3 \pm 2\sqrt{2} \]

### Step 5: Choose the positive root
Since \(x\) represents a power, we take the positive root:
\[ x = 3 + 2\sqrt{2} \]

### Step 6: Find \(\left(\frac{a}{b}\right)^{\frac{5}{7}}\)
We know that:
\[ \left(\frac{a}{b}\right)^{\frac{5}{7}} = \left(\left(\frac{a}{b}\right)^{\frac{5}{14}}\right)^2 = x^2 \]

### Step 7: Calculate \(x^2\)
Calculating \(x^2\):
\[ x^2 = (3 + 2\sqrt{2})^2 = 9 + 12\sqrt{2} + 8 = 17 + 12\sqrt{2} \]

### Step 8: Find \(\left(\frac{b}{a}\right)^{\frac{5}{7}}\)
Similarly, we have:
\[ \left(\frac{b}{a}\right)^{\frac{5}{7}} = \left(\frac{1}{x}\right)^{\frac{5}{7}} = \frac{1}{x^2} \]

### Step 9: Calculate \(\frac{1}{x^2}\)
To find \(\frac{1}{x^2}\):
\[ \frac{1}{x^2} = \frac{1}{17 + 12\sqrt{2}} \]
To rationalize the denominator:
\[ \frac{1}{17 + 12\sqrt{2}} \cdot \frac{17 - 12\sqrt{2}}{17 - 12\sqrt{2}} = \frac{17 - 12\sqrt{2}}{(17 + 12\sqrt{2})(17 - 12\sqrt{2})} \]
Calculating the denominator:
\[ (17)^2 - (12\sqrt{2})^2 = 289 - 288 = 1 \]
Thus:
\[ \frac{1}{x^2} = 17 - 12\sqrt{2} \]

### Step 10: Combine the results
Now we can find:
\[ \left(\frac{a}{b}\right)^{\frac{5}{7}} + \left(\frac{b}{a}\right)^{\frac{5}{7}} = x^2 + \frac{1}{x^2} = (17 + 12\sqrt{2}) + (17 - 12\sqrt{2}) = 34 \]

### Final Answer
Thus, the value of \(\left(\frac{a}{b}\right)^{\frac{5}{7}} + \left(\frac{b}{a}\right)^{\frac{5}{7}} = 34\).

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