a and b are positive numbers that satisfy the equation \frac {1}{a} - \frac {1}{b} = \frac {1}{a+b} . Determine the value of \frac {a^6}{b^6} + \frac {b^6} {a^6} .
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To solve this problem, we first need to simplify the equation given and then find the values of a and b. Once we have the values of a and b, we can calculate the value of the expression \(\frac{a^6}{b^6} + \frac{b^6}{a^6}\).
Starting with the equation \(\frac{1}{a} - \frac{1}{b} = \frac{1}{a+b}\), we can simplify it by cross-multiplying:
\(b(\frac{1}{a} - \frac{1}{b}) = a+b\)
\(b - a = (a+b)\left(\frac{1}{a} - \frac{1}{b}\right)\)
Expanding the right side of the equation:
\(b - a = \frac{(a+b)}{a} - \frac{(a+b)}{b}\)
Next, we can find a common denominator:
\(b - a = \frac{b(a+b)}{ab} - \frac{a(a+b)}{ab}\)
Combining the fractions on the right side:
\(b - a = \frac{b(a+b) - a(a+b)}{ab}\)
Simplifying the numerator:
\(b - a = \frac{ab + b^2 - a^2 - ab}{ab}\)
Canceling out the common terms:
\(b - a = \frac{b^2 - a^2}{ab}\)
Factoring the numerator:
\(b - a = \frac{(b-a)(b+a)}{ab}\)
Now, we can cancel out the factor (b - a) on both sides:
\(1 = \frac{b+a}{ab}\)
Cross-multiplying again:
\(ab = b+a\)
Rearranging the terms:
\(ab - a - b = 0\)
Using the quadratic equation formula:
\(a = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(ab - a - b)}}{2}\)
Simplifying:
\(a = \frac{1 \pm \sqrt{1 + 4(a+b-ab)}}{2}\)
Next, we'll solve for b:
Similarly, we can rearrange the terms and use the quadratic equation formula for b:
\(b = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(ab - a - b)}}{2}\)
Simplifying:
\(b = \frac{1 \pm \sqrt{1 + 4(a+b-ab)}}{2}\)
Now that we have the values of a and b, we can calculate the value of the expression \(\frac{a^6}{b^6} + \frac{b^6}{a^6}\):
\(\frac{a^6}{b^6} + \frac{b^6}{a^6} = \left(\frac{a^6}{b^6} + \frac{b^6}{a^6}\right) \times \frac{a^6b^6}{a^6b^6}\)
Expanding the numerator:
\( = \frac{a^{12} + b^{12}}{a^6b^6}\)
This is the final value of the expression \(\frac{a^6}{b^6} + \frac{b^6}{a^6}\).