91Ó°ÊÓ

Understanding the role of coefficients in algebra is vital for mastering multiplication of monomials. Coefficients are the numerical part of a term, which when multiplied with a variable, define the term's value. In the given exercise, the coefficients are -\r\(\frac{1}{2}\) and -\r\(\frac{1}{4}\). When multiplying monomials, the first step always involves multiplying these coefficients together.

It's important to note that the sign of the coefficient affects the outcome. In this case, the multiplication of two negative coefficients results in a positive product. Therefore, -\r\(\frac{1}{2} \times -\frac{1}{4} = \frac{1}{8}\). This positive -\r\(\frac{1}{8}\) now becomes the coefficient of the resulting monomial after the multiplication process.

When dealing with variables and exponents, it's essential to grasp how they interact during multiplication. An exponent indicates how many times a variable is multiplied by itself. For example, -\r\(a^3\) means -\r\(a\) multiplied by itself three times. When multiplying monomials with the same base, such as -\r\(a^3 \times a^2\), you simply add the exponents. This is a shortcut based on the fundamental principle of exponents that states -\r\(a^m \times a^n = a^{m+n}\).

Thus, in our exercise, the correct approach is to combine the exponents of -\r\(a\) from both monomials by adding them: -\r\(3 + 2 = 5\), resulting in -\r\(a^5\). This step is crucial for simplifying and determining the power of the variable in the final answer.

To multiply monomials effectively, certain steps must be followed systematically. We can break this process down into three manageable parts:

1. Multiply the Coefficients: Start with the numerical parts of each monomial. Remember that signs matter, and the multiplication rules for positive and negative numbers apply.
2. Multiply the Variables: For variables with exponents, you need to add the exponents together if the bases are the same, as discussed in the section on variables and exponents.
3. Combine the Results: The final monomial is formed by combining the product of the coefficients with the newly determined power of the variable.

Applying these steps to our original exercise, we multiplied the coefficients -\r\(\frac{1}{2}\) and -\r\(\frac{1}{4}\) to get -\r\(\frac{1}{8}\), then we added the exponents of -\r\(a\) to get -\r\(a^5\), and finally combined them to achieve the result -\r\(\frac{1}{8}a^5\). Clear application of these steps ensures a correct solution and builds a strong foundation for more complex algebraic operations.