In Exercises 1–8, multiply the monomials.(3x²)(5x⁴)

Verified step by step guidance

1. Identify the coefficients and the variables in each monomial: the first monomial is \(3x^2\) and the second is \(5x^4\).
2. Multiply the coefficients: \(3\) and \(5\).
3. Add the exponents of the like bases: \(x^2\) and \(x^4\).
4. Combine the results: the product of the coefficients and the sum of the exponents.
5. Write the final expression as a single monomial.

Key Concepts

Monomials

A monomial is a polynomial with only one term, which can be a constant, a variable, or a product of constants and variables raised to non-negative integer powers. In the expression (3x²)(5x⁴), both 3x² and 5x⁴ are monomials. Understanding monomials is essential for performing operations like multiplication.

Multiplication of Monomials

When multiplying monomials, you multiply the coefficients (numerical parts) and add the exponents of like bases (variables). For example, in (3x²)(5x⁴), you multiply 3 and 5 to get 15, and for the variable x, you add the exponents 2 and 4 to get x^(2+4) = x⁶. This rule simplifies the multiplication process.

Exponent Rules

Exponent rules are mathematical guidelines that dictate how to handle operations involving powers. The key rule for multiplication states that when multiplying like bases, you add the exponents. This is crucial for simplifying expressions involving variables raised to powers, as seen in the multiplication of the monomials in the question.