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Simplifying fractions is an essential concept in algebra. It helps make your calculations easier and your answers simpler to read. To simplify a fraction, you reduce it to its lowest terms by canceling out common factors in the numerator and denominator.

Imagine you have the fraction \(\frac{4}{8}\). Both 4 and 8 can be divided by the common factor 4. When you divide both, you get \(\frac{1}{2}\).

In the given exercise, we simplified \(\frac{(x-3)^2}{6x} \times \frac{x^2}{x-3}\) by first canceling out one \(x-3\) from the numerator and the denominator. This step is crucial because it makes the calculation straightforward and helps in reducing the complexity of the problem.

Understanding common factors is key to simplifying fractions and solving algebraic problems. A common factor is a number or variable that divides both the numerator and the denominator without leaving a remainder.

For example, in the given problem, \(x-3\) is a common factor in the numerator and the denominator of the expression \(\frac{(x-3)^2 \times x^2}{6x \times (x-3)}\). By canceling out a common factor, the fraction becomes simpler.

Let's break it down: First, we identify \(x-3\) as a common factor and cancel it out. Then, we are left with \((x-3) x^2 / 6x\). Often, finding and canceling common factors is like looking for patterns or similarities in algebraic expressions. It helps make equations easier to manage and solve.

Multiplying fractions is straightforward but requires careful attention to detail, especially when variables are involved. To multiply fractions, multiply the numerators together and the denominators together.

In our exercise, we dealt with the expression \(\frac{(x-3)^2}{6x} \times \frac{x^2}{x-3}\). First, we performed fraction multiplication by flipping the second fraction and then multiplying the fractions directly. This process transformed the problem into \frac{(x-3)^2 x^2}{6x (x-3)}\.

After multiplication, it’s important to simplify further, which may involve canceling out common factors, as in this example. Mastering multiplication and the subsequent simplification of fractions is foundational for more complex algebraic manipulations and problem-solving.