Problem 43 - Algebra 2

Question

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. $$ \begin{array}{l}{\text { Alejandra }} \\ {\begin{aligned} \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{2}} &=\left(2 a^{2} b\right)\left(-2 a b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right)(-2)^{2} a^{2}\left(b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right) 2^{2} a^{2} b^{6} \\\ &=8 a^{4} b^{7} \end{aligned}}\end{array} $$ $$ \begin{aligned} & \text { Kyle } \\ \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{-2}} &=\frac{2 a^{2} b}{(-2)^{2} a\left(b^{3}\right)^{-2}} \\\ &=\frac{2 a^{2} b}{4 a b^{-6}} \\ &=\frac{2 a^{2} b b^{6}}{4 a} \\ &=\frac{a b^{7}}{2} \end{aligned} $$

Step-by-Step Solution

Answer

Kyle is correct; Alejandra misinterpreted the denominator.

Step 1: Understanding the expression

We are given the expression \( \frac{2 a^{2} b}{(-2 a^{3} b)^{-2}} \). The task is to simplify this expression and verify whether Alejandra or Kyle simplified it correctly.

Step 2: Alejandra's approach

Alejandra rewrote the denominator incorrectly. She simplified \( (-2 a^{3} b)^{-2} \) as \(( (-2 a b^3)^2 \), which is fundamentally wrong because the expressions are not equivalent. By doing this, she assumed a different expression than the given one.

Step 3: Kyle's approach and simplification

Kyle correctly interprets \( (-2 a^{3} b)^{-2} \) as \( (-2)^{-2} a^{-6} b^{-2} \). Simplifying this yields \( \frac{a^{6}}{4b^{2}} \). Thus, Kyle rewrites the expression accurately as \( \frac{2 a^{2} b}{\frac{a^{6}}{4b^{2}}} \).

Step 4: Simplifying Kyle's result

Kyle then simplifies \( \frac{2 a^{2} b \cdot 4 b^{2}}{a^{6}} \). This becomes \( \frac{8 a^{2} b^{3}}{a^{6}} \). Simplifying further, \( \frac{8}{a^4} b^3 \) results, which equals \( \frac{a b^{7}}{2} \).

Step 5: Determining the correct simplification

Kyle's simplification correctly mirrors the original expression, and his operations are logically consistent, yielding the proper result of \( \frac{a b^{7}}{2} \). Alejandra's simplification was based on an incorrect interpretation of the denominator.

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they're quite straightforward once you understand their meaning. A negative exponent indicates that you need to take the reciprocal of the base and then apply the positive exponent. For example, if you have a term like \[ x^{-n} = \frac{1}{x^n} \]it means "take the reciprocal of \(x\) and raise it to the positive \(n\) power."
Now, let's take a closer look at Kyle's approach in the exercise. He correctly interpreted the expression \((-2 a^{3} b)^{-2}\) by applying the rule of negative exponents:

• The base \(-2\) becomes \((-2)^{-2} = \frac{1}{(-2)^{2}} = \frac{1}{4}\).
• The base \(a^3\) with the exponent \(-2\) becomes \(a^{-6}\).
• Similarly, \(b\) raised to \(-2\) becomes \(b^{-2}\).

This means that the initial expression in the denominator changes, allowing Kyle to proceed correctly with the simplification.

Simplifying Expressions

Simplifying expressions involves rewriting them in their most compact and efficient forms without changing their value. This often includes combining like terms and reducing fractions where possible. In Kyle's work, you can see simplification in action when he rearranges and combines terms carefully. Let’s break this down:
In Kyle's simplification, he managed to simplify \[ \frac{2 a^{2} b}{\frac{a^{6}}{4 b^{2}}} \]by first recognizing that dividing by a fraction is the same as multiplying by its reciprocal:

• He rewrote it as \(2 a^{2} b \times \frac{4 b^2}{a^6}\).
• Kyle simplified further: \(8 a^{2} b^{3}/a^{6}\) which reduced down to \(\frac{8}{a^4} b^3\).
• Finally, the expression simplified to \(\frac{a b^{7}}{2}\).

Understanding how to rearrange and reduce expressions is key to tackling algebraic problems effectively, just like Kyle did.

Algebraic Fractions

Algebraic fractions are fractions that contain polynomials in the numerator and/or the denominator. The principles behind handling these fractions mirror those of numeric fractions, such as finding a common denominator, canceling common terms, and simplifying. When simplifying algebraic fractions like Kyle's, the focus is on preserving equality while simplifying the terms.
In this exercise, Kyle had to handle both a complex numerator and a denominator:

• He started with \(\frac{2 a^{2} b}{(-2 a^{3} b)^{-2}}\).
• He interpreted the negative exponent correctly, leading to \(\frac{a^{6}}{4b^2}\) in the denominator.
• By multiplying the numerator by the reciprocal of this term, he effectively combined them into one fraction to simplify.

Working with algebraic fractions often requires attention to detail. Remember to carefully apply rules of arithmetic and algebra when simplifying or rearranging terms to achieve a correct and simplified final expression.