In Exercises 1–14, multiply using the product rule.(5x³y⁴)(20x⁷y⁸)

Verified step by step guidance

1. Identify the expression to be multiplied: \((5x^3y^4)(20x^7y^8)\).
2. Apply the product rule for exponents, which states that when multiplying like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
3. Multiply the coefficients: \(5 \times 20\).
4. Add the exponents for the \(x\) terms: \(x^3 \cdot x^7 = x^{3+7}\).
5. Add the exponents for the \(y\) terms: \(y^4 \cdot y^8 = y^{4+8}\).

Key Concepts

Product Rule

The product rule is a fundamental property of exponents that states when multiplying two powers with the same base, you add their exponents. For example, a^m * a^n = a^(m+n). This rule is essential for simplifying expressions involving variables raised to powers, allowing for efficient calculations in algebra.

Multiplying Monomials

Multiplying monomials involves multiplying the coefficients (numerical parts) and applying the product rule to the variables. For instance, when multiplying (5x³y⁴) and (20x⁷y⁸), you first multiply the coefficients (5 * 20) and then combine the variables by adding their exponents according to the product rule.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. In the context of multiplying monomials, this concept is crucial for ensuring that the final expression is in its simplest form, making it easier to interpret and use in further calculations.