Simplify Square Root of 250: A Step-by-Step Guide

To simplify the square root of 250, we will follow these steps:

1. Factor 250 into its prime factors: \(250 = 2 \times 5^2 \times 5\)
2. Rewrite the square root of the product: \(\sqrt{250} = \sqrt{5^2 \times 10}\)
3. Separate the square roots: \(\sqrt{5^2 \times 10} = \sqrt{5^2} \times \sqrt{10}\)
4. Simplify the square root of 5²: \(\sqrt{5^2} = 5\)
5. Combine the results: \(\sqrt{250} = 5\sqrt{10}\)

Therefore, the square root of 250 simplified is:

In decimal form, this is approximately:

\(\sqrt{250} \approx 15.81139\)

Example Problem

Let's apply this knowledge to a practical example:

James wants to buy a new rug for his living room. He finds a square rug that has an area of 250 square feet. To find the length of each side of the rug, we need to take the square root of 250:

The length of each side of the rug is approximately 15.81 feet.

Additional Information

The square root is the inverse operation of squaring. It can be found using various methods such as prime factorization and the long division method.

Interactive Questions

• Is the square root of 250 equal to 10 times the square root of 25?
• What is the square root of 250 in simplest radical form?

Answering these questions helps reinforce the concept of simplifying square roots.

The process of simplifying the square root of 250 involves breaking it down into its prime factors and finding the simplest radical form. The square root of 250 can be expressed as 5√10. This result is derived by factoring 250 into 25 and 10, then taking the square root of 25 and simplifying. Understanding these steps helps in simplifying similar square roots.

1. Factor 250 into its prime components: 250 = 25 × 10
2. Take the square root of the perfect square factor: √25 = 5
3. Combine the results: 5√10

This method not only simplifies calculations but also aids in comprehending the principles behind square roots.

The concept of square roots is fundamental in mathematics. A square root of a number \( x \) is a value \( y \) such that \( y^2 = x \). For example, the square root of 250 is a number that, when squared, gives 250.

To simplify the square root of 250, we follow these steps:

1. Find the prime factorization of 250: \( 250 = 2 \times 5^3 \).
2. Identify the pairs of prime factors: \( 250 = (5^2) \times (5 \times 2) \).
3. Take the square root of the perfect square: \( \sqrt{5^2} = 5 \).
4. Combine the results: \( \sqrt{250} = 5\sqrt{10} \).

Therefore, the simplified form of \( \sqrt{250} \) is \( 5\sqrt{10} \).

This process of simplifying square roots helps in various mathematical computations and makes complex problems easier to solve.

Simplifying the square root of 250 involves breaking down the number into its prime factors and simplifying the radical expression. Below are the steps to simplify √250:

1. List the Factors: Begin by listing the factors of 250. These are: 1, 2, 5, 10, 25, 50, 125, and 250.
2. Identify the Perfect Squares: From the list of factors, identify the perfect squares. For 250, the perfect square is 25 (since 25 = 5²).
3. Divide by the Largest Perfect Square: Divide 250 by the largest perfect square identified, which is 25. This gives us:
   250 ÷ 25 = 10
4. Take the Square Root of the Perfect Square: Calculate the square root of the largest perfect square. For 25, it is 5:
   √25 = 5
5. Simplify the Radical: Combine the results to simplify the square root of 250. Since we have 25 as the perfect square and 10 as the remaining factor, we get:
   √250 = √(25 × 10) = √25 × √10 = 5√10

Thus, the square root of 250 simplified is 5√10.

Simplifying square roots is a fundamental skill in algebra that involves breaking down a number into its prime factors and simplifying the expression. Here are some methods to simplify square roots effectively:

• Prime Factorization: Factor the number under the square root into its prime factors and group the pairs of prime numbers.
• Using the Rule of Radicals: Apply the rule \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) to simplify the square root by splitting it into smaller, more manageable parts.
• Perfect Squares: Identify and extract perfect square factors to simplify the square root. For example, \( \sqrt{250} = \sqrt{25 \times 10} = \sqrt{25} \times \sqrt{10} = 5\sqrt{10} \).

These methods help in breaking down complex square roots into simpler forms, making them easier to work with in various mathematical problems.

Understanding how to simplify the square root of 250 can be easier with some practical examples. Here are a few step-by-step examples to help illustrate the process:

1. Example 1: Simplifying √250

   1. Identify the prime factors of 250. The factors are 2, 5, and 5.
   2. Rewrite 250 as the product of its prime factors: 250 = 5² × 10.
   3. Take the square root of each factor: √250 = √(5² × 10).
   4. Since √(5²) = 5, the expression simplifies to 5√10.
   5. Thus, √250 in simplest form is 5√10.

2. Example 2: Using the Simplified Form

   If you have a rug with an area of 250 square feet, and you want to find the length of one side of the rug:

   • The area of the rug is 250 sq ft, so each side is √250 ft.
   • Using the simplified form, √250 = 5√10.
   • The approximate decimal value is 15.81 ft, so each side of the rug is about 15.81 feet long.

Here are some frequently asked questions about simplifying the square root of 250:

• What is the square root of 250 simplified?

The simplified form of the square root of 250 is \( 5\sqrt{10} \). This is because 250 can be factored into \( 25 \times 10 \), and the square root of 25 is 5, leaving the square root of 10 inside the radical.

• Is the square root of 250 a rational or irrational number?

The square root of 250 is an irrational number because it cannot be expressed as a simple fraction.

• What is the decimal form of the square root of 250?

The decimal form of the square root of 250 is approximately 15.8114.

• How do you find the square root of 250 using the long division method?

The long division method involves pairing the digits of the number, finding the closest square root, and iterating through division and averaging steps to get a more precise value. This method is useful for manual calculations without a calculator.

• Is the square root of 250 a real number?

Yes, the square root of 250 is a real number as it exists on the number line.

To assist you in simplifying the square root of 250 and other mathematical expressions, here are some interactive tools you can use:

• Mathway Square Root Calculator:

  This tool allows you to find the square root of any given number. Simply enter the number, and it will calculate the square root for you, providing both exact and decimal forms. It also offers step-by-step explanations.

• Omni Calculator's Square Root Calculator:

  Omni Calculator provides a square root calculator that simplifies square roots by identifying factors and perfect squares. It offers detailed explanations and examples to help you understand the process.

• Simplify Calculator by Mathway:

  This tool is designed to simplify complex expressions, including those with square roots. Enter your expression, and the calculator will reduce it to its simplest form with detailed steps.

• Symbolab Simplifying Radicals Calculator:

  Symbolab's calculator helps in simplifying radicals. It breaks down the steps involved in the simplification process and provides detailed solutions for better understanding.

These tools are highly recommended for anyone looking to simplify square roots or other mathematical expressions efficiently. They provide detailed explanations and step-by-step solutions to enhance your learning experience.

Simplifying the square root of 250 is an important mathematical skill that showcases the process of breaking down a number into its prime factors and identifying perfect squares. Through methods such as prime factorization and the long division method, you can see that √250 simplifies to 5√10. This simplification not only makes calculations easier but also deepens your understanding of the relationships between numbers.

The key steps involved in simplifying square roots include:

• Identifying the prime factors of the number.
• Grouping the factors into pairs.
• Extracting and simplifying the perfect squares.
• Combining the results to get the simplified form.

For √250, the steps show that it can be expressed as 5√10, which is approximately 15.81 in decimal form. Understanding these steps and practicing with different numbers can enhance your problem-solving skills and mathematical fluency.

In summary, whether using the prime factorization method or the long division method, simplifying square roots is a valuable technique that can be applied across various mathematical problems. Keep practicing and exploring different numbers to master this fundamental concept.