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

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Identify the coefficients (numerical parts) and the variables with their exponents separately in the expression \((-3m^{4})(6m^{2})(-4m^{5})\).

Multiply the coefficients together: \(-3 \times 6 \times -4\). Remember that multiplying two negative numbers results in a positive number.

Apply the product rule of exponents for the variable \(m\): when multiplying like bases, add their exponents. So, add the exponents \(4 + 2 + 5\).

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.

Double-check the signs and exponents to ensure the expression is fully simplified and correctly written.

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주요 개념

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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.