Let's simplify the expression:

$\frac{1}{\left(\frac{1}{b} + \frac{1}{a}\right)}$

Step 1: Combine the terms in the denominator.

To combine the terms $\frac{1}{b}$ and $\frac{1}{a}$, you need a common denominator. The least common denominator (LCD) of $b$ and $a$ is $ab$.

$\frac{1}{b} + \frac{1}{a} = \frac{a}{ab} + \frac{b}{ab} = \frac{a + b}{ab}$

Step 2: Substitute back into the original expression.

Now the expression becomes:

$\frac{1}{\left(\frac{a + b}{ab}\right)}$

Step 3: Simplify the complex fraction.

To simplify this, multiply by the reciprocal of the denominator:

$\frac{1}{\left(\frac{a + b}{ab}\right)} = \frac{ab}{a + b}$

Final simplified form:

$\frac{ab}{a + b}$

So, the simplified expression is:

$\frac{ab}{a + b}$

Would you like further explanation or any clarifications on this? Here are 5 related questions you might find interesting:

1. How do we simplify more complex fractions with multiple terms in the numerator and denominator?
2. What would happen if we had different constants or variables in the denominator, like $x$ and $y$?
3. How does this formula relate to the harmonic mean of two numbers?
4. How can you solve for $a$ or $b$ if this expression is equal to a constant?
5. Can you generalize this method for more than two terms in the denominator?

Tip: When dealing with complex fractions, always try to find a common denominator to simplify them before proceeding with the overall expression!