$$ \frac{a}{b} $$$$ +\frac{b}{a} $$$$  $$$$ =1 $$ then $$  $$$$ a^3+b^3= $$

We are given the equation:

\[
\frac{a}{b} + \frac{b}{a} = 1
\]

First, let's simplify the left-hand side:

\[
\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}
\]

Thus, we now have the equation:

\[
\frac{a^2 + b^2}{ab} = 1
\]

Multiply both sides of this equation by $ ab $:

\[
a^2 + b^2 = ab
\]

We are asked to find $ a^3 + b^3 $. We can use the identity for the sum of cubes:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

Since we already know that $ a^2 + b^2 = ab $, we can substitute $ ab $ for $ a^2 + b^2 $:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (a + b)(0) = 0
\]

Thus, $ a^3 + b^3 = 0 $.

Would you like further details on any part of this solution? 

Here are 5 related questions:
1. What happens if $ \frac{a}{b} + \frac{b}{a} \neq 1 $?
2. How can we generalize this approach for other values of the sum?
3. Can we solve for specific values of $ a $ and $ b $?
4. What if $ a $ and $ b $ are complex numbers?
5. What is the geometric interpretation of this result?

**Tip**: Whenever you see an expression like $ a^3 + b^3 $, remember the identity $ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $, which simplifies calculations.

If $$ \frac{a}{b} + \frac{b}{a} = 1 $$, then what is $$ a^3 + b^3 $$?

Explore how to solve the equation $ \frac{a}{b} + \frac{b}{a} = 1 $ and find the value of $ a^3 + b^3 $ using the sum of cubes identity. Learn step-by-step methods with algebraic simplifications, suitable for high school students in Grades 9-12.

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