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When subtracting fractions, it's essential to have a common denominator. This helps us combine the fractions easily. A denominator is the bottom number of a fraction, and it shows how many equal parts the item is divided into.

To find a common denominator, we look for the least common multiple (LCM) of the denominators involved. The LCM is the smallest number that is a multiple of both denominators. In the example \(-\frac{3}{4} - \frac{5}{8}\), the denominators are 4 and 8. The LCM of 4 and 8 is 8.

This means that we must convert any fractions with a denominator of 4 to one with a denominator of 8 before subtracting. If both fractions have the same denominator, you can easily subtract them.

Improper fractions are fractions where the numerator (the top number) is larger than the denominator (the bottom number). Improper fractions represent numbers greater than or equal to one.

For example, in this exercise, after performing the subtraction, we get \(-\frac{11}{8}\). This is an improper fraction because 11 is greater than 8. Such fractions can also be expressed as mixed numbers (a whole number and a proper fraction), but in this context, it's often simpler to leave them as improper fractions, especially when performing operations like subtraction.

Subtracting fractions involves several clear steps:

• Find a common denominator: Identify the least common multiple of the denominators.
• Convert the fractions: Adjust the fractions to have the same denominator. This usually involves multiplying both the numerator and denominator by the same number.
• Perform the subtraction: Subtract the numerators while keeping the common denominator the same.
• Simplify the result: If the resulting fraction can be simplified, do so.

In our example, we followed these steps:

1. The common denominator for 4 and 8 is 8.
2. Convert \-\frac{3}{4}\ to \-\frac{6}{8}\.
3. Subtract: \-\frac{6}{8} - \frac{5}{8} = -\frac{11}{8}\.
4. The result \(-\frac{11}{8}\) is already simplified.