Simplify each expression. See Example 1. (-3m⁴) (6m²) (-4m⁵)

Verified step by step guidance

1. Identify the coefficients (numerical parts) and the variables with their exponents separately in the expression \((-3m^{4})(6m^{2})(-4m^{5})\).
2. Multiply the coefficients together: \(-3 \times 6 \times -4\). Remember that multiplying two negative numbers results in a positive number.
3. Apply the product rule of exponents for the variable \(m\): when multiplying like bases, add their exponents. So, add the exponents \$4 + 2 + 5$.
4. Combine the results from the coefficient multiplication and the variable with the new exponent to write the simplified expression in the form \(a m^{b}\), where \(a\) is the product of coefficients and \(b\) is the sum of exponents.
5. Double-check the signs and exponents to ensure the expression is fully simplified and correctly written.

Key Concepts

Multiplication of Coefficients

When multiplying algebraic expressions, multiply the numerical coefficients (constants) separately from the variables. For example, in (-3)(6)(-4), multiply the numbers first to get the product of the coefficients.

Laws of Exponents

When multiplying variables with the same base, add their exponents. For instance, m⁴ × m² × m⁵ equals m^(4+2+5) = m¹¹. This rule simplifies expressions involving powers of the same variable.

Handling Negative Signs in Multiplication

Multiplying negative numbers follows the rule that an even number of negative factors results in a positive product, while an odd number results in a negative product. This helps determine the overall sign of the simplified expression.