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Understanding the product of powers property is essential when multiplying variables with the same base. This property states that when you multiply two expressions with the same base, you can add the exponents. For example, if you have expressions like \( x^a \times x^b \), the result is \( x^{a+b} \). This is because you are effectively combining the number of times the base is used as a factor.

In the exercise \((-4x^3)(3x^2)\), after handling the coefficients separately, apply the product of powers property to the variables. Here the base is \( x \) in both cases, and the exponents are 3 and 2, respectively. By adding them, \( 3+2 \), you conclude that \( x^5 \) is the simplified expression for the variable part.

• Always ensure the bases are the same before adding exponents.
• This property simplifies expressions and is a key step in polynomial multiplication.
• It's helpful for maintaining order and accuracy.

The process of multiplying coefficients is straightforward and involves basic arithmetic. Coefficients are the numerical parts of the terms in an algebraic expression. In our case, they are -4 and 3 from the terms \(-4x^3\) and \(3x^2\). Simply multiply these two numbers together.

By solving \(-4 \times 3\), you get \(-12\). This result represents the new coefficient in the product of the two polynomials.

Remember:

• Consider the signs of numbers: a positive times a negative is always negative.
• This step is vital for simplifying any expression and reducing errors.
• Always write down results neatly and in steps to avoid mistakes.

These basics are essential, especially when dealing with larger polynomials.

Exponent rules involve understanding how powers or exponents work in multiplication and division of variables. These rules exist to simplify complex expressions and make calculations manageable.

The main rule applied in this exercise is the product of powers rule, which is a specific case of exponent rules. Here, you add the exponents when multiplying variables with the same base, as with \( x^3 \) and \( x^2 \), resulting in \( x^{3+2} = x^5 \).

Understanding these rules is essential:

• They simplify expressions by reducing the number of variables and terms.
• Exponents can represent repeated multiplication in an easy-to-manage format.
• Knowing when and how to apply these rules can solve even the most complex algebraic problems.

Mastering these rules ensures efficiency and accuracy in solving algebraic equations.