In Exercises 1–14, multiply using the product rule.(-3x⁴y⁰z)(-7xyz³)

Verified step by step guidance

1. Identify the terms to be multiplied: \((-3x^4y^0z)\) and \((-7xyz^3)\).
2. Apply the product rule for exponents, which states that when multiplying like bases, you add the exponents: \(x^a \cdot x^b = x^{a+b}\).
3. Multiply the coefficients: \(-3\) and \(-7\) to get \(21\).
4. Multiply the \(x\) terms: \(x^4\) and \(x^1\) to get \(x^{4+1} = x^5\).
5. Multiply the \(z\) terms: \(z^1\) and \(z^3\) to get \(z^{1+3} = z^4\).

Key Concepts

Product Rule

The product rule is a fundamental property in algebra that states when multiplying two expressions, you can multiply their coefficients and add the exponents of like bases. This rule simplifies the process of multiplying polynomials and other algebraic expressions, ensuring that the final result is expressed in its simplest form.

Exponents

Exponents represent the number of times a base is multiplied by itself. In the context of multiplication, when two terms with the same base are multiplied, their exponents are added together. Understanding how to manipulate exponents is crucial for simplifying expressions and solving equations in algebra.

Coefficients

Coefficients are the numerical factors in a term of an algebraic expression. When multiplying terms, the coefficients are multiplied together to form the new coefficient of the resulting term. Recognizing and correctly handling coefficients is essential for accurate calculations in algebraic operations.