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When working with monomials, one of the first tasks is to simplify the coefficients. Coefficients are the numerical parts of the terms. In our example, we begin with \(\frac{-30}{-36} \). The first thing to note is that both numbers are negative, so they cancel each other out. This means we can simplify it to \(\frac{30}{36} \).

To make this fraction as simple as possible, we find the Greatest Common Divisor (GCD) of 30 and 36. The GCD is the largest number that can divide both numbers without leaving a remainder. For 30 and 36, the GCD is 6.

So, we divide both the numerator and the denominator by 6:
\[\begin{equation} \ \frac{30 \div 6}{36 \div 6} = \ \frac{5}{6} \ \ \end{equation}\]
This leaves us with the simplified coefficient 5/6.

• Step 1: Identify the coefficients.
• Step 2: Determine the GCD.
• Step 3: Divide the coefficients by the GCD.

The Greatest Common Divisor (GCD) is a crucial concept when simplifying monomials. It's the largest number that can evenly divide both the numerator and the denominator. Knowing how to find the GCD can make your work much easier.

To find the GCD of two numbers, you can use the following process:

• List the factors of each number.
• Find the largest factor that appears in both lists.

For instance, let's find the GCD of 30 and 36:
30: 1, 2, 3, 5, 6, 10, 15, 30
36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The largest number in both lists is 6, making it the GCD.

Finding the GCD helps simplify numbers easily, making it an essential skill in prealgebra.

Negative exponents can seem tricky, but they are quite manageable when you understand the rules. In our example, we have \(\frac{x^8}{x^9} \). To simplify this, use the rule for dividing powers with the same base: subtract the exponents.

\[\begin{equation} \ \frac{x^8}{x^9} = x^{8-9} = x^{-1} \ \ \end{equation}\]
A negative exponent indicates the reciprocal of the base raised to the positive of that exponent.

So, \ x^{-1} \ becomes \ \frac{1}{x} \ .

The rule simplifies division problems and helps maintain clear results. Remember:

• Subtract the exponents when dividing.
• Change negative exponents to positive by taking the reciprocal.

These steps give us a simplified result that combines the coefficient and variable parts.