Simplify each expression. See Example 1. (5x²y) (-3x³y⁴)

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Identify the expression to simplify: \((5x^{2}y)(-3x^{3}y^{4})\).

Apply the associative property of multiplication to group the coefficients and the variables separately: \((5 \times -3)(x^{2} \times x^{3})(y \times y^{4})\).

Multiply the coefficients: \(5 \times -3 = -15\).

Use the product of powers property for the variables with the same base: \(x^{2} \times x^{3} = x^{2+3} = x^{5}\) and \(y \times y^{4} = y^{1+4} = y^{5}\).

Combine all parts to write the simplified expression: \(-15x^{5}y^{5}\).

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Multiplication of Monomials

Multiplying monomials involves multiplying their coefficients (numerical parts) and then applying the laws of exponents to variables with the same base. For example, (5x²y) × (-3x³y⁴) requires multiplying 5 and -3, then combining powers of x and y.

Laws of Exponents

When multiplying variables with the same base, add their exponents. For instance, x² × x³ equals x^(2+3) = x⁵. This rule applies to all variables involved in the expression to simplify powers correctly.

Handling Negative Coefficients

When multiplying coefficients, consider their signs. Multiplying a positive number by a negative number results in a negative product. In the example, 5 × (-3) equals -15, which affects the overall sign of the simplified expression.