What Is The Simplest Form Of The Expression For The Side Length, In Feet?A. 4 50 4 \sqrt{50} 4 50 B. 5 6 5 \sqrt{6} 5 6 C. 10 2 10 \sqrt{2} 10 2 D. 20 2 20 \sqrt{2} 20 2

Mar 8, 2025 by ADMIN 178 views

What is the simplest form of the expression for the side length, in feet?

Introduction

When dealing with geometric shapes, particularly triangles, it's essential to understand how to calculate the side lengths. In this article, we will explore the concept of side length and how to simplify expressions involving square roots. We will examine a specific problem and determine the simplest form of the expression for the side length in feet.

Understanding the Problem

The problem involves finding the simplest form of an expression for the side length of a triangle. The given options are:

A. $4 \sqrt{50}$ B. $5 \sqrt{6}$ C. $10 \sqrt{2}$ D. $20 \sqrt{2}$

To determine the correct answer, we need to understand the concept of side length and how to simplify expressions involving square roots.

Concept of Side Length

The side length of a triangle is the distance between two vertices. In a right-angled triangle, the side lengths can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Simplifying Expressions Involving Square Roots

To simplify an expression involving a square root, we need to find the largest perfect square that divides the number inside the square root. We can then rewrite the expression as the product of the square root of the perfect square and the remaining number.

Example Problem

Let's consider the expression $4 \sqrt{50}$ . To simplify this expression, we need to find the largest perfect square that divides 50. We can rewrite 50 as $25 \times 2$ , where 25 is a perfect square.

Step 1: Factorize 50

$50 = 25 \times 2$

Step 2: Rewrite the expression

$4 \sqrt{50} = 4 \sqrt{25 \times 2}$

Step 3: Simplify the expression

$4 \sqrt{25 \times 2} = 4 \times 5 \times \sqrt{2}$

Step 4: Simplify further

$4 \times 5 \times \sqrt{2} = 20 \times \sqrt{2}$

Conclusion

The simplified form of the expression $4 \sqrt{50}$ is $20 \sqrt{2}$ .

Comparison with Options

Now that we have simplified the expression, let's compare it with the given options:

A. $4 \sqrt{50}$ B. $5 \sqrt{6}$ C. $10 \sqrt{2}$ D. $20 \sqrt{2}$

The simplified expression $20 \sqrt{2}$ matches option D.

Final Answer

The simplest form of the expression for the side length, in feet, is $20 \sqrt{2}$ .

Importance of Simplifying Expressions

Simplifying expressions involving square roots is essential in mathematics, particularly in geometry and trigonometry. It helps to make calculations easier and more efficient. In this article, we have demonstrated how to simplify an expression involving a square root and determined the simplest form of the expression for the side length in feet.

Real-World Applications

Simplifying expressions involving square roots has numerous real-world applications. For example, in architecture, engineers use mathematical calculations to design buildings and bridges. Simplifying expressions involving square roots can help them make accurate calculations and ensure the stability of the structure.

Conclusion

In conclusion, simplifying expressions involving square roots is a crucial concept in mathematics. By understanding how to simplify expressions, we can make calculations easier and more efficient. In this article, we have demonstrated how to simplify an expression involving a square root and determined the simplest form of the expression for the side length in feet.

Introduction

In our previous article, we explored the concept of simplifying expressions involving square roots and determined the simplest form of the expression for the side length in feet. In this article, we will address some frequently asked questions (FAQs) on this topic.

Q1: What is the purpose of simplifying expressions involving square roots?

A1: The purpose of simplifying expressions involving square roots is to make calculations easier and more efficient. By simplifying expressions, we can reduce the complexity of mathematical problems and make it easier to solve them.

Q2: How do I simplify an expression involving a square root?

A2: To simplify an expression involving a square root, you need to find the largest perfect square that divides the number inside the square root. You can then rewrite the expression as the product of the square root of the perfect square and the remaining number.

Q3: What is a perfect square?

A3: A perfect square is a number that can be expressed as the product of an integer and itself. For example, 4 is a perfect square because it can be expressed as 2 x 2.

Q4: How do I find the largest perfect square that divides a number?

A4: To find the largest perfect square that divides a number, you can factorize the number into its prime factors. You can then identify the perfect square factors and rewrite the expression accordingly.

Q5: Can I simplify expressions involving square roots using a calculator?

A5: Yes, you can simplify expressions involving square roots using a calculator. However, it's essential to understand the underlying mathematical concepts to ensure that you are using the calculator correctly.

Q6: What are some common mistakes to avoid when simplifying expressions involving square roots?

A6: Some common mistakes to avoid when simplifying expressions involving square roots include:

Not identifying the largest perfect square factor
Not rewriting the expression correctly
Not simplifying the expression fully

Q7: How do I check if my simplified expression is correct?

A7: To check if your simplified expression is correct, you can plug it back into the original equation and verify that it is true. You can also use a calculator to check the expression.

Q8: Can I simplify expressions involving square roots with negative numbers?

A8: Yes, you can simplify expressions involving square roots with negative numbers. However, you need to be careful when dealing with negative numbers, as they can affect the sign of the expression.

Q9: How do I simplify expressions involving square roots with fractions?

A9: To simplify expressions involving square roots with fractions, you need to find the largest perfect square that divides the numerator and denominator separately. You can then rewrite the expression accordingly.

Q10: Can I simplify expressions involving square roots with decimals?

A10: Yes, you can simplify expressions involving square roots with decimals. However, you need to be careful when dealing with decimals, as they can affect the accuracy of the expression.

Conclusion

In conclusion, simplifying expressions involving square roots is a crucial concept in mathematics. By understanding how to simplify expressions, we can make calculations easier and more efficient. In this article, we have addressed some frequently asked questions (FAQs) on this topic and provided guidance on how to simplify expressions involving square roots.

Additional Resources

For further learning, we recommend the following resources:

Khan Academy: Simplifying Square Roots
Mathway: Simplifying Square Roots
Wolfram Alpha: Simplifying Square Roots

Final Thoughts

Simplifying expressions involving square roots is a fundamental concept in mathematics. By mastering this concept, you can make calculations easier and more efficient. Remember to always identify the largest perfect square factor, rewrite the expression correctly, and simplify the expression fully. With practice and patience, you can become proficient in simplifying expressions involving square roots.