In Exercises 1–8, multiply the monomials.(3x²y⁴)(5xy⁷)

Verified step by step guidance

1. Identify the coefficients and variables in each monomial: (3x^2y^4) and (5xy^7).
2. Multiply the coefficients: 3 * 5.
3. Apply the product of powers property to the x terms: x^2 * x^1 = x^(2+1).
4. Apply the product of powers property to the y terms: y^4 * y^7 = y^(4+7).
5. Combine the results to form the final expression: (3 * 5)x^(2+1)y^(4+7).

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 (3x²y⁴)(5xy⁷), both 3x²y⁴ and 5xy⁷ 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 variables. For example, in (3x²y⁴)(5xy⁷), you multiply 3 and 5 to get 15, and for the variables, you add the exponents of x (2 + 1) and y (4 + 7) to find the new exponents. This process is crucial for simplifying the expression.

Exponent Rules

Exponent rules govern how to handle powers of numbers and variables during multiplication and division. The key rules include the product of powers (adding exponents) and the power of a product (distributing exponents). Mastery of these rules is vital for correctly simplifying expressions involving monomials, such as in the given multiplication problem.