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Multiplying fractions is a straightforward process in fraction arithmetic. To multiply two fractions, follow these simple steps:

• Multiply the numerators (the top numbers) together to get the new numerator.
• Multiply the denominators (the bottom numbers) together to get the new denominator.
• Simplify the resulting fraction if possible.

Let's illustrate this with our original exercise values. Given the fractions \( b = \frac{2}{3} \) and \( c = \frac{2}{9} \), we find their product as follows:

First, multiply the numerators: \( 2 \times 2 = 4 \).
Next, multiply the denominators: \( 3 \times 9 = 27 \).
So, \( b \times c = \frac{4}{27} \). This product is already in its simplest form since 4 and 27 have no common factors other than 1.

To subtract fractions, they must share the same denominator. When fractions have different denominators, we need to find a common denominator. Let's see how this works with the expression involving fractions \( a \) and \( b \cdot c \). Here are the steps to subtract these fractions:

• Identify the denominators of the fractions to be subtracted. In our exercise, \( a = \frac{5}{9} \) and \( b \cdot c = \frac{4}{27} \).
• Determine a common denominator if the denominators differ. The least common multiple (LCM) of the denominators is the smallest number that can be divided evenly by each original denominator.
• Convert each fraction to an equivalent fraction with the common denominator.
• Once they have the same denominator, subtract the numerators while keeping the denominator the same.

In the exercise, we start by identifying the LCM of 9 and 27, which is 27. Then, convert \( \frac{5}{9} \) so it has 27 as its denominator: \( \frac{5}{9} = \frac{15}{27} \). Finally, subtract \( \frac{15}{27} - \frac{4}{27} = \frac{11}{27} \). This gives us the solution in its simplest form.

Understanding the least common multiple (LCM) is crucial when working with fractions that need a common denominator. The LCM is the smallest positive integer that is divisible by two or more numbers.

Here’s how you find the LCM, particularly useful in fraction arithmetic:

• List the multiples of each number.
• Identify the smallest shared multiple from these lists.

For example, to find the LCM of 9 and 27, list the multiples:

- Multiples of 9: 9, 18, 27, 36, ...
- Multiples of 27: 27, 54, 81, ...

The smallest common multiple is 27. When subtracting fractions like \( \frac{5}{9} - \frac{4}{27} \), knowing the LCM of 9 and 27 allows us to adjust both fractions to have a denominator of 27, facilitating straightforward subtraction. Mastering the concept of LCM will simplify arithmetic operations involving fractions.