FIND THE ERROR. Alejandra and Kyle both simplified (2 a^2 b)/((-2 a^3 b)^-2) . Who is correct? E

step 1 In this problem, Alejandra and Kyle have solved this problem. 2a squared b over negative 2 a b c

FIND THE ERROR. Alejandra and Kyle both simplified (2 a^2...

FIND THE ERROR. Alejandra and Kyle both simplified $\frac{2 a^{2} b}{\left(-2 a^{3} b\right)^{-2}} .$ Who is correct? Explain your reasoning. $$ \begin{array}{l}{\text { Alejandra }} \\ {\begin{aligned} \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{2}} &=\left(2 a^{2} b\right)\left(-2 a b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right)(-2)^{2} a^{2}\left(b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right) 2^{2} a^{2} b^{6} \\ &=8 a^{4} b^{7} \end{aligned}}\end{array} $$ $$ \begin{aligned} & \text { Kyle } \\ \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{-2}} &=\frac{2 a^{2} b}{(-2)^{2} a\left(b^{3}\right)^{-2}} \\ &=\frac{2 a^{2} b}{4 a b^{-6}} \\ &=\frac{2 a^{2} b b^{6}}{4 a} \\ &=\frac{a b^{7}}{2} \end{aligned} $$

FIND THE ERROR. Alejandra and Kyle both simplified $\frac{2 a^{2} b}{\left(-2 a^{3} b\right)^{-2}} .$ Who is correct? Explain your reasoning.
$$
\begin{array}{l}{\text { Alejandra }} \\ {\begin{aligned} \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{2}} &=\left(2 a^{2} b\right)\left(-2 a b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right)(-2)^{2} a^{2}\left(b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right) 2^{2} a^{2} b^{6} \\ &=8 a^{4} b^{7} \end{aligned}}\end{array}
$$
$$
\begin{aligned} & \text { Kyle } \\ \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{-2}} &=\frac{2 a^{2} b}{(-2)^{2} a\left(b^{3}\right)^{-2}} \\ &=\frac{2 a^{2} b}{4 a b^{-6}} \\ &=\frac{2 a^{2} b b^{6}}{4 a} \\ &=\frac{a b^{7}}{2} \end{aligned}
$$

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Key Concepts

Exponentiation of a Product

When an entire product is raised to an exponent, the exponent must be applied to every factor inside the parentheses. For example, in an expression like (-2a^3b)^(2), the exponent is distributed to -2, a^3, and b separately, resulting in (-2)^2 * (a^3)^2 * b^2. Missing or misapplying this distribution is a common source of error when simplifying such expressions.

Negative Exponents

A negative exponent indicates that the base is on the opposite side of the fraction line. Specifically, for any nonzero element x and exponent n, x^(-n) is equivalent to 1/(x^n). Recognizing this rule is important when an expression in the denominator has a negative exponent, as it allows the expression to be correctly rewritten and simplified.

Order of Operations and Grouping

Properly following the order of operations is crucial in algebra. This includes correctly interpreting which terms are affected by the exponent, especially when parentheses are involved. A small misinterpretation in grouping—such as mixing up the exponents applied to different variables—can lead to errors in the simplified answer. Understanding and maintaining the intended grouping is essential for correct manipulation of algebraic expressions.

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