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

Verified step by step guidance

1. Identify the expression to simplify: \((5x^{2}y)(-3x^{3}y^{4})\).
2. Apply the multiplication property of exponents by multiplying the coefficients (numbers) together: multiply 5 and -3.
3. Use the product rule for exponents on the variable \(x\): when multiplying like bases, add the exponents, so \(x^{2} \times x^{3} = x^{2+3}\).
4. Use the product rule for exponents on the variable \(y\): multiply \(y\) (which is \(y^{1}\)) by \(y^{4}\), so \(y^{1} \times y^{4} = y^{1+4}\).
5. Combine the results from the coefficients and variables to write the simplified expression as a product of the new coefficient and the variables with their summed exponents.

Key Concepts

Multiplication of Monomials

When multiplying monomials, multiply their coefficients (numerical parts) and then multiply variables by adding their exponents if the bases are the same. For example, (5x²)(-3x³) equals -15x⁵.

Laws of Exponents

The laws of exponents state that when multiplying like bases, add their exponents. For instance, x² * x³ = x^(2+3) = x⁵. This rule applies to all variables with exponents in the expression.

Combining Variables with Different Exponents

When variables with the same base but different exponents are multiplied, add the exponents to simplify. For example, y * y⁴ = y^(1+4) = y⁵. This helps in simplifying expressions involving multiple variables.