How to Simplify the Square Root of 50

To simplify the square root of 50, follow these steps:

Prime Factorization Method

1. Factor 50 into its prime factors: \( 50 = 2 \times 5^2 \)
2. Rewrite the square root of 50 using these factors: \( \sqrt{50} = \sqrt{2 \times 5^2} \)
3. Since \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we get: \( \sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2} \)
4. Since \( \sqrt{5^2} = 5 \), the expression simplifies to: \( 5\sqrt{2} \)

Thus, the simplified form of \( \sqrt{50} \) is \( 5\sqrt{2} \).

Long Division Method

1. Start with the number 50. Place a bar over 50 and pair the digits in the decimal part.
2. Find the largest number whose square is less than or equal to 50. In this case, it's 7 because \( 7^2 = 49 \).
3. Subtract 49 from 50 to get a remainder of 1, then bring down pairs of zeros.
4. Double the quotient (7) to get 14. Now find a digit (X) such that \( 14X \times X \leq 100 \). Here, X is 0 because \( 140 \times 0 = 0 \).
5. Continue this process to get more decimal places.

Using this method, \( \sqrt{50} \approx 7.071 \).

Key Points

• The square root of 50 is not a perfect square.
• The simplified radical form is \( 5\sqrt{2} \).
• In decimal form, it is approximately 7.071.
• The square root of 50 is an irrational number.

Examples and Applications

Here are some practical applications of finding the square root of 50:

1. Fencing a Square Plot: If you have a square plot of 50 square feet, the side length would be \( \sqrt{50} \approx 7.071 \) feet. The perimeter would then be \( 4 \times 7.071 \approx 28.284 \) feet.
2. Travel Distance: If traveling at a speed of \( 5\sqrt{50} \approx 35.355 \) miles per hour for half an hour, the distance covered would be \( 35.355 \times 0.5 \approx 17.677 \) miles.

Conclusion

Simplifying the square root of 50 helps in various mathematical and practical applications. Whether using prime factorization or long division, understanding these methods provides a deeper insight into the properties of square roots.

• Introduction

• What is the Square Root of 50?

• Methods to Simplify the Square Root of 50

  1. Prime Factorization Method
  2. Long Division Method

• Simplified Form of the Square Root of 50

• Step-by-Step Simplification

  1. Prime Factorization Steps
  2. Long Division Steps

• Exact and Decimal Forms of the Square Root of 50

• Applications and Examples

  1. Example Problems
  2. Real-Life Applications

• FAQs on the Square Root of 50

The concept of square roots is fundamental in mathematics, where the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 50 can be expressed in several forms, including its simplified radical form, as a decimal, and in exponential notation. This section will guide you through the basic principles and methods used to simplify and understand square roots, focusing on the example of √50.

1. Definition of Square Roots: Understanding the basic concept of square roots and their significance in mathematics.
2. Perfect Squares: Exploring what perfect squares are and why 50 is not a perfect square.
3. Simplified Radical Form: How to simplify the square root of 50 to its radical form (5√2).
4. Decimal Form: Converting the square root of 50 to its approximate decimal form (7.071).
5. Exponential Notation: Representing the square root of 50 using exponents (50^1/2).
6. Methods to Find Square Roots: Detailed methods such as prime factorization and the long division method.
7. Applications: Practical applications of square roots in real-world problems.
8. Interactive Examples: Examples to solidify understanding through practice.

To simplify the square root of 50, we need to break it down into its prime factors and then simplify using the properties of square roots. Here's a detailed, step-by-step guide:

1. Identify the prime factors of 50:

   • 50 = 2 × 25
   • 25 = 5 × 5
   • So, 50 = 2 × 5 × 5

2. Write 50 as the product of its prime factors:

   \(\sqrt{50} = \sqrt{2 \times 5^2}\)

3. Apply the square root to each factor:

   \(\sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2}\)

4. Simplify the square root of the perfect square:

   \(\sqrt{5^2} = 5\)

5. Combine the simplified square roots:

   \(\sqrt{2} \times 5 = 5\sqrt{2}\)

Thus, the simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\). This method utilizes the prime factorization and the property that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).

Understanding the properties of square roots is crucial for simplifying and working with square roots effectively. Here are some important properties to consider:

• Non-Negative Property: The square root of a non-negative number is also non-negative. For any non-negative number \(a\), \(\sqrt{a} \geq 0\).
• Product Property: The square root of a product is equal to the product of the square roots of the factors. For any non-negative numbers \(a\) and \(b\), \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
• Quotient Property: The square root of a quotient is equal to the quotient of the square roots of the numerator and the denominator. For any non-negative numbers \(a\) and \(b\) (with \(b \neq 0\)), \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).
• Power Property: The square root of a power can be simplified by taking half of the exponent. For any non-negative number \(a\) and any integer \(n\), \(\sqrt{a^n} = a^{n/2}\).
• Addition and Subtraction: The square root of a sum or difference is not generally equal to the sum or difference of the square roots. In other words, \(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\) and \(\sqrt{a - b} \neq \sqrt{a} - \sqrt{b}\).
• Zero Property: The square root of zero is zero. \(\sqrt{0} = 0\).
• Square of a Square Root: The square of a square root returns the original number. For any non-negative number \(a\), \((\sqrt{a})^2 = a\).

By using these properties, we can simplify and manipulate square roots more effectively. Let's apply some of these properties to the square root of 50:

• Prime factorization of 50: \(50 = 2 \cdot 5^2\)
• Using the product property: \(\sqrt{50} = \sqrt{2 \cdot 25} = \sqrt{2} \cdot \sqrt{25}\)
• Since \(\sqrt{25} = 5\): \(\sqrt{50} = \sqrt{2} \cdot 5 = 5\sqrt{2}\)

The square root of 50, approximately 7.071, finds its application in various fields due to its mathematical properties. Here are some notable applications:

• Geometry and Construction:

  In geometry, the square root of 50 can be used to determine the length of the diagonal of a square with sides of 5 units. The diagonal \( d \) of a square is given by the formula:

  \[ d = a\sqrt{2} \]

  For a square with side length 5 units:

  \[ d = 5\sqrt{2} = \sqrt{50} \approx 7.071 \text{ units} \]

• Physics:

  In physics, the square root of 50 can be used in problems involving velocity, acceleration, and distance. For instance, if an object moves with a uniform acceleration and covers a distance of 50 units in a certain time, the relation between these quantities often involves the square root of 50.

• Engineering:

  In engineering, particularly in electrical and mechanical fields, root values like the square root of 50 are used in formulas for calculating loads, forces, and electrical properties. For example, the calculation of RMS (Root Mean Square) values in AC circuits might involve square roots.

• Statistics:

  In statistics, the square root of 50 is used when dealing with standard deviations and variances. For a dataset where the variance sums to 50, the standard deviation would be the square root of 50, providing insight into the data's spread.

• Real-life Problems:

  For practical applications, consider a scenario where you need to calculate the fencing required for a square plot of land that has an area of 50 square units. The side length of the plot would be the square root of 50, which helps in determining the amount of fencing material needed:

  \[ \text{Side length} = \sqrt{50} \approx 7.071 \text{ units} \]

  \[ \text{Perimeter} = 4 \times 7.071 \approx 28.284 \text{ units} \]

• Travel and Navigation:

  When calculating distances using Pythagorean theorem in navigation, the square root of 50 can be involved. For example, if you need to travel 5 units north and then 5 units east, the straight-line distance would be the hypotenuse of a right triangle with legs of 5 units each:

  \[ \text{Distance} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} \approx 7.071 \text{ units} \]

• What is the square root of 50?

  The square root of 50 is represented as √50. In its simplified form, it is 5√2, which is approximately equal to 7.071.

• How do you simplify the square root of 50?

  To simplify the square root of 50, factorize 50 into its prime factors:

  • 50 = 2 × 5 × 5
  • √50 = √(2 × 5²) = 5√2
• Is the square root of 50 a rational number?

  No, the square root of 50 is an irrational number because it cannot be expressed as a simple fraction and its decimal representation is non-terminating and non-repeating.

• What is the decimal form of the square root of 50?

  The decimal form of the square root of 50 is approximately 7.071.

• What is the exponent form of the square root of 50?

  The exponent form of the square root of 50 is 50^1/2.

• Can the square root of 50 be simplified further?

  No, the square root of 50 in its simplest radical form is 5√2 and cannot be simplified further.

• What are the properties of the square root of 50?

  The square root of 50 is a non-perfect square, a quadratic surd, and an irrational number.

Below are several examples and practice problems to help you understand how to simplify the square root of 50 and apply it in various contexts.

Example 1: Simplifying the Square Root of 50

To simplify \(\sqrt{50}\) , follow these steps:

1. Prime factorize the number 50: \(50 = 2 \times 5^2\)
2. Express the square root using these factors: \(\sqrt{50} = \sqrt{2 \times 5^2}\)
3. Separate the square root of the perfect square: \(\sqrt{2 \times 5^2} = 5\sqrt{2}\)

Thus, the simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\) .

Example 2: Application in Geometry

Kevin wants to buy a square plot with an area of 50 square feet. To find the side length of the plot:

1. Use the formula for the area of a square: \(\text{Area} = \text{side}^2\)
2. Solve for the side length: \(\text{side} = \sqrt{\text{Area}} = \sqrt{50} = 5\sqrt{2}\)

Thus, each side of the square plot is \(5\sqrt{2}\) feet long.

Practice Problem 1

Simplify the square root of 72.

Solution:

1. Prime factorize the number 72: \(72 = 2^3 \times 3^2\)
2. Express the square root using these factors: \(\sqrt{72} = \sqrt{2^3 \times 3^2}\)
3. Separate the square root of the perfect squares: \(\sqrt{2^3 \times 3^2} = 3\sqrt{8} = 3\sqrt{4 \times 2} = 3 \times 2\sqrt{2} = 6\sqrt{2}\)

Thus, the simplified form of \(\sqrt{72}\) is \(6\sqrt{2}\) .

Practice Problem 2

If Randal is traveling at an average speed of \(5\sqrt{50}\) miles per hour for half an hour, how far does he travel?

Solution:

1. Calculate the speed: \(5\sqrt{50} = 5 \times 5\sqrt{2} = 25\sqrt{2}\) miles per hour
2. Use the distance formula: \(\text{Distance} = \text{Speed} \times \text{Time}\)
3. Substitute the values: \(\text{Distance} = 25\sqrt{2} \times 0.5 = 12.5\sqrt{2}\)

Therefore, Randal travels approximately \(12.5\sqrt{2}\) miles.

Practice Problem 3

Find the square root of 98 and simplify it.

Solution:

1. Prime factorize the number 98: \(98 = 2 \times 7^2\)
2. Express the square root using these factors: \(\sqrt{98} = \sqrt{2 \times 7^2}\)
3. Separate the square root of the perfect squares: \(\sqrt{2 \times 7^2} = 7\sqrt{2}\)

Thus, the simplified form of \(\sqrt{98}\) is \(7\sqrt{2}\) .

Practice Problem 4

Simplify the square root of 200.

Solution:

1. Prime factorize the number 200: \(200 = 2^3 \times 5^2\)
2. Express the square root using these factors: \(\sqrt{200} = \sqrt{2^3 \times 5^2}\)
3. Separate the square root of the perfect squares: \(\sqrt{2^3 \times 5^2} = 5\sqrt{8} = 5\sqrt{4 \times 2} = 5 \times 2\sqrt{2} = 10\sqrt{2}\)

Thus, the simplified form of \(\sqrt{200}\) is \(10\sqrt{2}\) .