Calculating The Perimeter Of A Rectangle With Radical Dimensions

In the realm of mathematics, geometry introduces us to a myriad of shapes, each possessing unique properties and characteristics. Among these shapes, the rectangle stands out as a fundamental figure, defined by its four sides and four right angles. Determining the perimeter of a rectangle, the total distance around its boundary, is a common mathematical task. However, when the dimensions of a rectangle involve radicals, the process requires careful simplification and manipulation of these expressions. This article delves into the intricacies of calculating the perimeter of a rectangle with dimensions expressed as radicals, providing a comprehensive guide to the steps involved.

Deconstructing the Dimensions: Simplifying Radicals

To embark on our journey of perimeter calculation, we first need to unravel the dimensions of the rectangle, which are presented as radicals: $\sqrt{50 a^3 b^2}$ and $\sqrt{200 a^3}$. These radicals, while seemingly complex, can be simplified by factoring out perfect squares from under the radical sign. Let's break down each dimension:

Simplifying $\sqrt{50 a^3 b^2}$

The radical $\sqrt{50 a^3 b^2}$ can be simplified by identifying perfect square factors within the expression. We can express 50 as $25 \cdot 2$, where 25 is a perfect square. Similarly, $a^3$ can be written as $a^2 \cdot a$, where $a^2$ is a perfect square, and $b^2$ is already a perfect square. Applying these factorizations, we get:

$\sqrt{50 a^3 b^2} = \sqrt{25 \cdot 2 \cdot a^2 \cdot a \cdot b^2}$

Now, we can extract the square roots of the perfect squares:

$\sqrt{25 \cdot 2 \cdot a^2 \cdot a \cdot b^2} = 5ab\sqrt{2a}$

Therefore, the simplified form of the first dimension is $5ab\sqrt{2a}$. This meticulous simplification is a crucial step in accurately calculating the perimeter.

Simplifying $\sqrt{200 a^3}$

Following a similar approach, we simplify the second dimension, $\sqrt{200 a^3}$. We can express 200 as $100 \cdot 2$, where 100 is a perfect square, and $a^3$ as $a^2 \cdot a$, where $a^2$ is a perfect square. This leads us to:

$\sqrt{200 a^3} = \sqrt{100 \cdot 2 \cdot a^2 \cdot a}$

Extracting the square roots of the perfect squares, we obtain:

$\sqrt{100 \cdot 2 \cdot a^2 \cdot a} = 10a\sqrt{2a}$

Thus, the simplified form of the second dimension is $10a\sqrt{2a}$. With both dimensions now in their simplest radical forms, we can proceed to calculate the perimeter with greater ease.

Unveiling the Perimeter: Calculation and Simplification

The perimeter of a rectangle is defined as the total length of its sides, which can be calculated using the formula: Perimeter = 2(length + width). In our case, the length and width are represented by the simplified radical expressions we derived earlier.

Applying the Perimeter Formula

Substituting the simplified dimensions into the perimeter formula, we get:

Perimeter = $2(5ab\sqrt{2a} + 10a\sqrt{2a})$

Distributive Property

To further simplify the expression, we apply the distributive property, multiplying the 2 by each term inside the parentheses:

Perimeter = $2 \cdot 5ab\sqrt{2a} + 2 \cdot 10a\sqrt{2a}$

This simplifies to:

Perimeter = $10ab\sqrt{2a} + 20a\sqrt{2a}$

Combining Like Terms

Now, we observe that both terms in the expression contain the radical $\sqrt{2a}$. These are considered like terms, and we can combine them by adding their coefficients:

Perimeter = $(10ab + 20a)\sqrt{2a}$

Final Simplified Expression

Therefore, the simplified expression for the perimeter of the rectangle is $(10ab + 20a)\sqrt{2a}$. This expression represents the total distance around the rectangle in its most concise form.

Student's Solution: A Critical Analysis

The problem statement presents a student's attempt to calculate the perimeter. Let's examine the student's work and identify any potential errors.

The student's solution is as follows:

$\begin{aligned} 2 \sqrt{50 a^3 b^2}+2 \sqrt{200 a^3} & =2 \cdot 5 a b \sqrt{2 a}+2 \cdot 10 a \sqrt{2 a} \\ & =10 a b \sqrt{2 a}+20 a\sqrt{2a} \end{aligned}$

Identifying the Discrepancy

Upon closer inspection, we notice a subtle but significant difference between the student's final answer and our derived solution. The student correctly simplifies the radicals and applies the distributive property, arriving at the expression $10 a b \sqrt{2 a}+20 a\sqrt{2a}$. However, the student stops at this point and fails to combine the like terms.

The Missing Step: Combining Like Terms

The key step that the student overlooked is the combination of the like terms, $10ab\sqrt{2a}$ and $20a\sqrt{2a}$. Both terms share the same radical factor, $\sqrt{2a}$, making them eligible for combination. By adding their coefficients, we arrive at the correct simplified expression: $(10ab + 20a)\sqrt{2a}$.

The Significance of Simplification

This analysis highlights the importance of complete simplification in mathematical problems. While the student demonstrated a good understanding of radical simplification and the distributive property, the failure to combine like terms resulted in an incomplete answer. The ability to recognize and combine like terms is a fundamental skill in algebra and is crucial for obtaining the most accurate and concise solution.

Key Concepts and Takeaways

This exploration of rectangle perimeter calculation with radical dimensions reinforces several key mathematical concepts:

• Simplifying Radicals: The ability to identify and extract perfect square factors from under the radical sign is essential for simplifying radical expressions.
• Perimeter Formula: The perimeter of a rectangle is calculated using the formula: Perimeter = 2(length + width).
• Distributive Property: The distributive property allows us to multiply a factor across terms within parentheses, a crucial step in simplifying expressions.
• Combining Like Terms: Recognizing and combining like terms, those with the same variable and radical factors, is essential for obtaining the most simplified expression.
• Attention to Detail: Mathematical accuracy hinges on meticulous attention to detail, ensuring that each step is executed correctly and no simplification opportunities are missed.

Conclusion: Mastering Radical Dimensions

Calculating the perimeter of a rectangle with radical dimensions requires a systematic approach, encompassing radical simplification, application of the perimeter formula, distributive property, and the combination of like terms. By mastering these techniques, we can confidently navigate mathematical challenges involving radicals and geometric shapes. This exploration serves as a testament to the power of mathematical tools in unraveling complex problems and arriving at elegant solutions. The journey from initial radical expressions to the final simplified perimeter underscores the beauty and precision inherent in mathematics, where each step builds upon the previous one, leading to a comprehensive understanding of the problem at hand.