Problem 48 - Algebra 2

Question

REVIEW. Four students worked the same math problem. Each student's work is shown below. $$ \begin{array}{ll}{\frac{\text { Student } \mathrm{F}}{x^{2} x^{-5}=\frac{x^{2}}{x^{5}}}} & {\frac{\text { Student } \mathrm{G}}{x^{2} x^{-5}=\frac{x^{2}}{x^{-5}}}} \\ {=\frac{1}{x^{3}}, x \neq 0} & {=x^{7}, x \neq 0}\end{array} $$ $$ \begin{array}{l}{\frac{\text { Student } \mathrm{H}}{x^{2} x^{-5}=\frac{x^{2}}{x^{-5}}}} \\ {\quad=x^{-7}, x \neq 0}\end{array} $$ $$ \begin{array}{l}{\text { Student I }} \\ {\begin{aligned} x^{2} x^{-5} &=\frac{x^{2}}{x^{5}} \\ &=x^{3}, x \neq 0 \end{aligned}}\end{array} $$ Which is a completely correct solution? $$ \begin{array}{ll}{\mathbf{F} \text { Student } \mathrm{F}} & {\mathbf{H} \text { Student } \mathrm{H}} \\ {\text { G Student } \mathrm{G}} & {\text { J Student } \mathrm{J}}\end{array} $$

Step-by-Step Solution

Answer

Student F has the completely correct solution.

Step 1: Identify the Given Problem

The problem is given to different students to simplify the expression \( x^2 \times x^{-5} \). The operation involves multiplying two powers of \( x \). The expected product follows the rules of exponents.

Step 2: Apply the Rules of Exponents

Recall the rule \( a^m \times a^n = a^{m+n} \). Applying this rule to the expression \( x^2 \times x^{-5} \), we add the exponents: \( 2 + (-5) \).

Step 3: Perform the Addition of Exponents

Calculate \( 2 + (-5) = -3 \). Therefore, the product of \( x^2 \times x^{-5} \) is \( x^{-3} \).

Step 4: Simplify the Expression

Convert \( x^{-3} \) to a fraction: \( \frac{1}{x^3} \) when \( x eq 0 \), to express the final solution in its simplest form.

Step 5: Compare Each Student's Solution to the Correct Answer

Student F: \( \frac{1}{x^3} \), which matches the correct solution. Student G: \( x^7 \), incorrect \( x^{-3} \) doesn't simplify to \( x^7 \). Student H: \( x^{-7} \), incorrect. Student I: \( x^3 \), incorrect since the exponents should be added, yielding \( x^{-3} \).

Step 6: Conclusion

The completely correct solution is by Student F, who provided the correct simplification \( \frac{1}{x^3} \).

Key Concepts

Simplification of Expressions

Simplifying expressions often involves rewriting them in a form that's easier to understand or use. One common technique in algebra is using the properties of exponents.

When you see terms like \( x^2 \times x^{-5} \), it involves a straightforward application of exponent rules. Remember, the rule \( a^m \times a^n = a^{m+n} \) states that when you multiply two powers with the same base, you simply add their exponents together.

In our exercise, \( x^2 \) and \( x^{-5} \) are such terms. Here's how to simplify:

• Add the exponents: \( 2 + (-5) \).
• This results in \( x^{-3} \).

The negative exponent indicates that \( x \) should be in the denominator, leading us to the simpler form, \( \frac{1}{x^3} \), for values of \( x \) that are not zero.

Algebra Problem-Solving

Tackling algebra problems often starts with understanding what is being asked and what is given. In the exercise, students were asked to simplify an expression, applying rules of exponents.

The systematic approach to solving algebra problems generally involves:

• Identifying the problem.
• Recognizing applicable mathematical rules or concepts.
• Applying these rules to transform the expression into its simplest form.

Here, the students needed to apply the rule of exponents correctly.

Student F successfully showed the capability to identify the need for this rule, apply it, and reach a conclusion that demonstrated mastery of basic algebraic techniques.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and making connections between known principles and the problem at hand. It’s crucial for deriving solutions correctly.

In our case, each student demonstrated different levels of reasoning when simplifying the expression \( x^2 \times x^{-5} \).

• Student G made a logical error in the exponent operation, demonstrating a misunderstanding of the exponent rule.
• Student H similarly misapplied the rule, resulting in a miscalculation.
• Student I ended up with \( x^{3} \), showing another typical mistake where negative exponents might be misinterpreted.

Understanding why Student F’s result \( \frac{1}{x^3} \) is correct involves recognizing that negative exponents reflect division in the context of simplifying expressions. Logical thinking in math ensures that operations like adding and subtracting exponents are executed correctly, leading to accurate results.