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When multiplying fractions, it's helpful to understand the structure of a fraction. Each fraction consists of a numerator (the top part) and a denominator (the bottom part).

To multiply fractions, you follow these steps:

• Multiply the numerators together.
• Multiply the denominators together.
• The resulting fraction is your product.

Let's look back at the exercise: \(\frac{3c}{d^2} \times \frac{8d}{6c^3} = \frac{3c \times 8d}{d^2 \times 6c^3}\).
We multiplied the numerators (3c and 8d) and got \(\text{24cd}\) and the denominators (d² and 6c³) and got \(\text{6c³d²}\).

Simplifying fractions means reducing them to their simplest form.
This is done by dividing the numerator and denominator by their greatest common divisor (GCD).
In our example fraction \(\frac{24cd}{6c^3d^2}\), we need to identify and divide out common factors in the numerator and denominator.
First, simplify the coefficients (24 and 6) by their GCD, which is 6.
So, \(\frac{24}{6} = 4\). Now our fraction looks like \(\frac{4cd}{c^3d^2}\).
Next, for the variables:\(c\) appears once in the numerator and three times in the denominator, and \(d\) appears once in the numerator and twice in the denominator.
Divide both top and bottom by \(c \times d\) and you get \(\frac{4}{c^2 \times d}\).
So, \(\frac{24cd}{6c^3d^2} -> \frac{4}{c^2 \times d}\).

The greatest common divisor (GCD) is the largest number that divides two or more numbers with no remainder.
Finding the GCD is crucial when simplifying fractions.
In our multiplication problem, we needed to reduce \(\frac{24cd}{6c^3d^2}\).
We looked at the numerical part first (24 and 6). The GCD of 24 and 6 is 6 because 6 divides both 24 and 6.
Then, for variables: \(c\) in the numerator and \(c^3\) in the denominator have \(c\) as the common factor, and \(d\) in the numerator and \(d^2\) in the denominator have \(d\) as the common factor.

• So, combined, the GCD of the entire expression is 6cd.
• This is why, after dividing everything by the GCD, we get a much simpler fraction.