91Ó°ÊÓ

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Algebraic expressions can be simple like \(3x\), or more complex like \(2x^3 + 5x^2 - x + 7\). They are used to represent real-world situations in a mathematical format.

In the exercise, we have the monomials \(-6 c^{4}\) and \(-12 c\). Monomials are a type of algebraic expression that consist of only one term.

An important aspect of working with algebraic expressions is the ability to combine like terms and perform operations, such as addition, subtraction, and multiplication. In this case, we are focusing on multiplying the monomials.

Coefficients are the numerical parts of terms in algebraic expressions. They are the numbers that multiply the variables. For example, in the term \(5x^2\), the coefficient is \(5\).

In the given exercise, we are multiplying the coefficients of the monomials \(-6 c^{4}\) and \(-12 c\). The coefficients here are \(-6\) and \(-12\), respectively.
To multiply the coefficients, we simply multiply the numbers together: \[ -6 \times -12 = 72 \]
The product of \(-6\) and \(-12\) is \(72\). Notice that a negative multiplied by a negative gives a positive result.

Exponents are used in algebraic expressions to denote repeated multiplication of a base by itself. For instance, in \(x^3\), the \(^3\) here represents the exponent, indicating that \(x\) is multiplied by itself three times: \(x \times x \times x\).

In the exercise, we are multiplying the variables with the same base, which is \(c\). The variables are \(c^4\) and \(c\). To do this, you add the exponents. Note that if a variable does not appear to have an exponent, it actually has an exponent of \(1\):
\[ c^4 \times c^1 = c^{4+1} = c^5 \]
This rule holds for any variables with the same base: you add their exponents together. Combining the results of the multiplied coefficients and the new exponent gives us our final answer:

\[72 c^5\]