Ch. R - Algebra Review

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. R - Algebra Review

Problem R.3.27

Chapter 1, Problem R.3.27

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

Verified step by step guidance

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.

Simplify the term inside the parentheses by replacing $n^{0}$ with 1, so the expression inside becomes $4m^{3} \times 1 = 4m^{3}$.

Apply the exponent of 2 to each factor inside the parentheses separately, using the rule $(ab)^{c} = a^{c}b^{c}$. This gives $(4)^{2} (m^{3})^{2}$.

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}$.

Don't forget the negative sign outside the parentheses. Multiply the result by $-1$ to get the simplified expression $-16m^{6}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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