Problem 78 - Algebra 2

Question

Error Analysis Explain why the following simplification is incorrect. \(5\left(4-5^{\frac{1}{2}}\right)=5(4)-5\left(5^{\frac{1}{2}}\right)=20-25^{\frac{1}{2}}=15\)

Step-by-Step Solution

Answer

The error lies in the incorrect application of the distributive property. Instead of correctly distributing over subtraction as \(20 - 5 \cdot (5^{\frac{1}{2}})\), the given simplification incorrectly shows it as \(20 - 25^{\frac{1}{2}}\). The correct answer should be 15.

Step 1: Identify the mistake

Looking at the given equation, it's clear that while distributing \(5\), a mistake was made. \(5\left(4-5^{\frac{1}{2}}\right)\) was incorrectly simplified to \(5(4)-5\left(5^{\frac{1}{2}}\right)=20-25^{\frac{1}{2}}=15\).

Step 2: Apply correct distributive property

The correct application of the distributive property is \(5\left(4-5^{\frac{1}{2}}\right)\), which simplifies to \(20 - 5 \cdot (5^{\frac{1}{2}})\) not \(20 - 25^{\frac{1}{2}}\). Here lies the root of the error. The term being subtracted should actually be \((5 \cdot (5^{\frac{1}{2}}))\), not \((25^{\frac{1}{2}})\). The exponent applies only to the \(5\) in the parenthesis, not the entire term including the leading \(5\).

Step 3: Correct final calculation

After the correct simplification with the distributive property, the correct final calculation should be \(20 - 5 \cdot (5^{\frac{1}{2}})\), which simplifies to \(20 - 5 \cdot 5 = 20 - 5 = 15\).

Key Concepts

Error Analysis

When analyzing an algebraic solution, it's crucial to understand where and why mistakes occur. Error analysis in mathematics helps us identify incorrect assumptions or misapplications of rules. Here, the primary mistake happened during the distribution process. A distinction must be made between properly distributing the multiplication over subtraction inside the parentheses.
In the expression \(5(4 - 5^{\frac{1}{2}})\), the error was extending the exponent mistakenly to the entire product \(5(5^{\frac{1}{2}})\), resulting instead in \(25^{\frac{1}{2}}\). Recognizing such errors allows us to correct them and apply rules accurately in the future. Practice in analyzing each step critically is vital in mathematics. It's more than arriving at the right answer—it's about understanding the journey there.

Simplification Mistakes

Simplification is a common source of mistakes in algebra. This error often happens when students rush through the process or overlook order of operations. In our primary example, the error was assuming that distributing the \(5\) affected the terms in the wrong way, and misunderstanding how the exponent affected the term it was specific to.
To avoid simplification mistakes, ensure that each step of simplification follows logically and sequentially correct use of operations. Remember:

• Distribute multiplication over addition or subtraction properly.
• Apply exponents accurately only to intended base numbers.
• Never change the operations incorrectly between stages of simplification.

Refining these techniques helps in maintaining accuracy and precision in solving algebraic problems, ensuring the computation matches the symbolic meanings intended.

Exponents in Algebra

Exponents represent repeated multiplication of a base number by itself. They are key in algebra for simplifying mathematical expressions. In the exercise, an exponent mistake occurred, disrupting the expression's accuracy.
Understanding how to manipulate exponents correctly is crucial:

• Apply an exponent only to the number directly preceding it, unless parentheses dictate otherwise.
• Remember that distributing a number like \(5\) to an expression \((n^{m})\) without combining it incorrectly, preserves each variable's role.

Misapplying exponents can significantly alter the meaning and outcome of an expression, as seen here where \(5^{\frac{1}{2}}\) was mishandled. By keeping these principles in mind, students can effectively navigate through algebraic challenges involving exponents.