Expand and simplify:
$\left(3-\frac{3}{x}\right)^2$
To expand and simplify the expression \(\left(3 - \frac{3}{x}\right)^2\), we recognize it as the square of a binomial. A binomial of the form \((a - b)^2\) follows the algebraic identity:
\[(a - b)^2 = a^2 - 2ab + b^2\]
In this specific case, we identify \(a = 3\) and \(b = \frac{3}{x}\). The expression is well-defined as long as \(x
eq 0\), which is a constraint we must keep in mind throughout the simplification process.
Now, we substitute \(a = 3\) and \(b = \frac{3}{x}\) into the identity \(a^2 - 2ab + b^2\):
\[\left(3 - \frac{3}{x}\right)^2 = 3^2 - 2 \cdot 3 \cdot \left(\frac{3}{x}\right) + \left(\frac{3}{x}\right)^2\]
Next, we perform the arithmetic for each term:
Substituting the evaluated terms back into the identity, we get:
\[\left(3 - \frac{3}{x}\right)^2 = 9 - \frac{18}{x} + \frac{9}{x^2}\]
This expression is now fully expanded. We can verify that no further simplification is needed as the terms \(9\), \(-\frac{18}{x}\), and \(\frac{9}{x^2}\) are in their simplest forms. Therefore, the expanded form is \(\boxed{9 - \frac{18}{x} + \frac{9}{x^2}}\).
Expanding algebraic expressions is a foundational skill used in physics and engineering. For example, when calculating the power dissipated in a circuit where the voltage and resistance are related by a variable, you might encounter expressions involving squares of rational functions. By expanding these expressions, you can more easily analyze how the power changes with respect to the variable component. This simplifies the process of finding maximum or minimum power points in practical engineering designs.