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When simplifying fractions, the greatest common divisor (GCD) is essential. It is the largest positive integer that divides both a given numerator and denominator without leaving a remainder.

In our example, we have \(- \frac{78}{90x} \). To find the GCD of 78 and 90, we list their factors.

• Factors of 78: 1, 2, 3, 6, 13, 26, 39, 78
• Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Comparing these factors, the greatest common factor is 6.

Using the GCD allows us to reduce the fraction to its simplest form, helping make calculations more efficient and easier to understand.

Now let's dive into the terms *numerator* and *denominator*. In a fraction, the numerator is the top number. It represents how many parts of a whole are being considered. The denominator is the bottom number, signifying the total number of equal parts the whole is divided into.

For example, in \(- \frac{78}{90x} \), 78 is the numerator, and 90\(x\) is the denominator. The role of the denominator in algebraic fractions can be slightly different since it may contain variables like \(x\). These variables communicate additional multiplicative parts.To simplify such an algebraic fraction, both the numeric part and the variable part must be considered to determine the fraction's simplest form.

Simplifying algebraic fractions combines finding the greatest common divisor and correctly managing numerators and denominators.

The goal is to make the fraction as brief as possible without changing its value. Given \(- \frac{78}{90x} \), we use the greatest common divisor found, which in this case was 6, and divide both the numerator and the term coefficient in the denominator by it. This process helps us reach the fraction\(- \frac{13}{15x} \).

• Divide 78 by 6 to obtain 13.
• Divide 90 by 6 to obtain 15. The \(x\) remains as part of the term.

After simplifying, always ensure there are no remaining common factors between the numerator and denominator, apart from 1, confirming the fraction is simplest. Simplifying helps in achieving clarity in complex calculations and ensures accuracy in further algebraic manipulations.