Simplify each expression. See Example 1. (-4x⁵) (4x²)

Verified step by step guidance

1. Identify the properties of exponents and multiplication that apply. When multiplying terms with the same base, you multiply the coefficients and add the exponents of the variables.
2. Multiply the coefficients: here, multiply -4 and 4, which gives \(-4 \times 4\).
3. Apply the product rule for exponents to the variable part: since the bases are the same (both are \(x\)), add the exponents \(5\) and \(2\) using \(x^{5} \times x^{2} = x^{5+2}\).
4. Combine the results from the coefficient multiplication and the exponent addition to write the simplified expression as a product of the new coefficient and the variable with the summed exponent.
5. Write the final simplified expression in the form \(ax^{b}\), where \(a\) is the product of the coefficients and \(b\) is the sum of the exponents.

Key Concepts

Product of Powers Property

When multiplying expressions with the same base, add the exponents. For example, x^a * x^b = x^(a+b). This property simplifies expressions involving powers of variables.

Multiplication of Coefficients

Multiply the numerical coefficients separately from the variables. For instance, in (-4x⁵)(4x²), multiply -4 and 4 to get -16 before combining the variable parts.

Simplifying Algebraic Expressions

Combine like terms and apply arithmetic operations to rewrite expressions in simpler forms. This involves using exponent rules and basic multiplication to reduce complexity.