Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. -(4m³n⁰)²

Verified step by step guidance

1. Recognize that the expression is \(-(4m^{3}n^{0})^{2}\). Start by understanding the components inside the parentheses: \$4\( is a constant, \)m^{3}\( is a variable term, and \)n^{0}$ equals 1 since any nonzero number raised to the zero power is 1.
2. Simplify the term inside the parentheses by replacing \(n^{0}\) with 1, so the expression inside becomes \(4m^{3} \times 1 = 4m^{3}\).
3. Apply the exponent of 2 to each factor inside the parentheses separately, using the rule \((ab)^{c} = a^{c}b^{c}\). This gives \((4)^{2} (m^{3})^{2}\).
4. Simplify each part: \((4)^{2} = 16\) and \((m^{3})^{2} = m^{3 \times 2} = m^{6}\). So the expression inside the parentheses squared is \$16m^{6}$.
5. Don't forget the negative sign outside the parentheses. Multiply the result by \(-1\) to get the simplified expression \(-16m^{6}\).

Key Concepts

Exponentiation of a Product

When raising a product to a power, apply the exponent to each factor inside the parentheses separately. For example, (ab)^n = a^n * b^n. This rule helps simplify expressions like (4m³n⁰)² by distributing the exponent 2 to 4, m³, and n⁰ individually.

Zero Exponent Rule

Any nonzero number raised to the zero power equals 1. For instance, n⁰ = 1 if n ≠ 0. This simplifies terms like n⁰ in the expression, effectively removing the variable from the product.

Negative Sign and Exponentiation

A negative sign outside parentheses affects the entire expression. When an expression is preceded by a negative sign and raised to a power, the sign remains outside unless the exponent is odd and the negative is inside the parentheses. Here, the negative sign applies after squaring the expression.