Square Root of 55 Simplified Radical Form: The Ultimate Guide

The square root of 55 in simplified radical form is represented as:

\(\sqrt{55}\)

It can be further broken down into simpler components as follows:

• \(\sqrt{55} = \sqrt{5 \cdot 11}\)
• \(\sqrt{55} = \sqrt{5} \cdot \sqrt{11}\)

Therefore, the simplified radical form of \(\sqrt{55}\) is \(\sqrt{5 \cdot 11}\).

The square root of 55, denoted as \( \sqrt{55} \), is a number that, when multiplied by itself, results in 55. In mathematical terms, this means:

\[ \sqrt{55} = q \quad \text{such that} \quad q^2 = 55 \]

Calculating the square root of a number is a fundamental concept in mathematics, often involving methods such as prime factorization, long division, and the Babylonian method. The square root of 55 is not a whole number and therefore, it is not a perfect square. This makes \( \sqrt{55} \) an irrational number, which means it cannot be expressed as a simple fraction and its decimal form is non-terminating and non-repeating.

In its simplest form, the square root of 55 remains \( \sqrt{55} \) because it cannot be simplified further into a product of smaller radicals. Here, we will explore the methods to find and simplify the square root of 55, discuss its properties, and provide examples and practice problems to reinforce understanding.

To summarize, the square root of 55 is an interesting example of an irrational number that helps illustrate various techniques in simplifying and approximating square roots.

The square root of a number is a value that, when multiplied by itself, gives the original number. For the number 55, its square root is denoted as \( \sqrt{55} \). The square root symbol \( \sqrt{} \) is known as the radical sign, and the number under the radical sign is called the radicand.

Here are some basic properties of the square root of 55:

• Non-Perfect Square: Since 55 is not a perfect square, its square root is an irrational number. This means it cannot be expressed as a simple fraction and its decimal form is non-repeating and non-terminating.
• Prime Factors: The prime factorization of 55 is 5 and 11. Neither of these factors are perfect squares, which means \( \sqrt{55} \) cannot be simplified further.
• Decimal Form: The approximate value of \( \sqrt{55} \) in decimal form is 7.4162, rounded to four decimal places.
• Principal Square Root: By convention, the square root function returns the principal (positive) square root. Therefore, \( \sqrt{55} \) refers to the positive value.

Understanding these properties helps in recognizing that the square root of 55 is an irrational number and is already in its simplest form.

The square root of 55 is an irrational number, which means it cannot be expressed as a simple fraction and its decimal form is non-repeating and non-terminating. However, we can use several methods to approximate or understand its value.

Prime Factorization

To simplify the square root of 55 using prime factorization:

• List the factors of 55: 1, 5, 11, 55
• Identify the prime factors: 55 = 5 × 11

Since neither 5 nor 11 is a perfect square, the square root of 55 cannot be simplified further. Thus, it remains as √55.

Long Division Method

This method is used to find the decimal approximation of √55:

1. Pair the digits of 55 from right to left: 55.
2. Find the largest number whose square is less than or equal to 55. This is 7, since 7² = 49.
3. Subtract 49 from 55, getting 6. Bring down two zeros to get 600.
4. Double the quotient (7) to get 14, and determine how many times 140 (14 with a trial digit) goes into 600. The answer is 4, since 144 × 4 = 576.
5. Subtract 576 from 600, get 24, bring down two more zeros to get 2400, and repeat the process.

Using this method, the square root of 55 is approximately 7.4162.

Babylonian Method

The Babylonian method, also known as Heron's method, is an iterative method to approximate the square root:

1. Start with an initial guess, say 27.5 (55/2).
2. Calculate a new guess by averaging the initial guess and 55 divided by the initial guess: (27.5 + 55/27.5)/2 = 14.75.
3. Repeat the process with the new guess: (14.75 + 55/14.75)/2 = 9.2394.
4. Continue iterating until the change between guesses is less than a small threshold (e.g., 0.001).

After several iterations, the approximate value for √55 is 7.4162.

Each method provides a different way to understand and approximate the square root of 55. Although the prime factorization confirms that it cannot be simplified further, both the long division and Babylonian methods help us find a numerical approximation.

To understand why the square root of 55 (√55) is an irrational number, we need to explore the definitions of rational and irrational numbers.

Rational Numbers: A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. These numbers can be terminating or repeating decimals.

Irrational Numbers: An irrational number, on the other hand, cannot be written as a simple fraction. Irrational numbers have non-terminating and non-repeating decimal expansions.

Now, let's consider the square root of 55:

• √55 is not a perfect square. This means there are no two identical integers that multiply to give 55.
• The decimal expansion of √55 is approximately 7.416198487, which does not terminate or repeat.

To see why this makes √55 irrational, let's break it down:

1. Since 55 is not a perfect square, its square root is not an integer.
2. Attempting to express √55 as a fraction will not yield a finite or repeating decimal, essential properties of rational numbers.
3. Using a calculator, √55 ≈ 7.416198487… The decimal goes on forever without repeating, confirming it as non-terminating and non-repeating.

Therefore, because √55 cannot be exactly expressed as a fraction and has a non-terminating, non-repeating decimal expansion, it is classified as an irrational number.

To simplify the square root of 55, follow these detailed steps:

Prime Factors

1. First, determine the prime factors of 55. The prime factors are 5 and 11 because \(55 = 5 \times 11\).
2. Since neither 5 nor 11 is a perfect square, we cannot simplify the square root further using prime factorization.

Nearest Perfect Squares

Identify the nearest perfect squares around 55. The perfect squares near 55 are:

• 49 (which is \(7^2\))
• 64 (which is \(8^2\))

Since 55 is not a perfect square, it remains in its simplified form as \(\sqrt{55}\).

Decimal Approximation

While \(\sqrt{55}\) cannot be simplified further in radical form, it can be approximated in decimal form:

\(\sqrt{55} \approx 7.4162\)

Therefore, the simplified form of the square root of 55 remains \(\sqrt{55}\) because it does not contain any factors that are perfect squares.

Understanding how to work with the square root of 55 can be reinforced through examples and practice problems. Here, we will explore adding and multiplying square roots, along with some practice problems to test your knowledge.

Example 1: Adding Square Roots

When adding square roots, they must have the same radicand. Unfortunately, √55 does not simplify to a form where it can be added to other common square roots. However, we can demonstrate the addition of square roots with similar radicands:

Example: Simplify \( \sqrt{55} + \sqrt{55} \)

Since both terms have the same radicand (55), we can add them directly:

\[ \sqrt{55} + \sqrt{55} = 2\sqrt{55} \]

Example 2: Multiplying Square Roots

Multiplying square roots is straightforward since it follows the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \):

Example: Simplify \( \sqrt{55} \times \sqrt{55} \)

\[ \sqrt{55} \times \sqrt{55} = \sqrt{55 \times 55} = \sqrt{3025} = 55 \]

Practice Problems

1. Simplify \( 3\sqrt{55} + 2\sqrt{55} \)
2. Calculate \( \sqrt{11} \times \sqrt{5} \)
3. Simplify \( \sqrt{275} \)
4. Determine if \( \sqrt{55} \) is rational or irrational.

Solutions to Practice Problems

• Problem 1:

  
  \( 3\sqrt{55} + 2\sqrt{55} = (3 + 2)\sqrt{55} = 5\sqrt{55} \)

• Problem 2:

  
  \( \sqrt{11} \times \sqrt{5} = \sqrt{11 \times 5} = \sqrt{55} \)

• Problem 3:

  
  \( \sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11} \)

• Problem 4:

  
  \( \sqrt{55} \) is irrational because 55 is not a perfect square and cannot be expressed as a fraction of two integers.

Test your understanding of the square root of 55 and its properties with these interactive questions. Use the steps and methods you've learned to solve each problem. Check your answers with the solutions provided.

• Question 1: Simplify the square root of 55 using the prime factorization method.

  Solution:

  \( \sqrt{55} = \sqrt{5 \times 11} \)

  Since neither 5 nor 11 is a perfect square, the square root of 55 cannot be simplified further. Thus, the simplified form remains \( \sqrt{55} \).

• Question 2: Determine if \( \sqrt{55} \) is a rational or irrational number.

  Solution:

  A rational number can be expressed as the quotient of two integers. Since 55 is not a perfect square, \( \sqrt{55} \) cannot be expressed as a fraction of two integers, making it an irrational number.

• Question 3: Use the long division method to approximate the square root of 55 to two decimal places.

  Solution:

  1. Estimate: Find two numbers between which the square root of 55 lies. Since \( 7^2 = 49 \) and \( 8^2 = 64 \), \( \sqrt{55} \) is between 7 and 8.
  2. Divide: \( 55 \div 7 = 7.857 \)
  3. Average: \( \frac{7 + 7.857}{2} = 7.4285 \)
  4. Refine: \( \frac{55}{7.4285} \approx 7.4162 \)

  Thus, \( \sqrt{55} \approx 7.42 \) to two decimal places.

• Question 4: Express the square root of 55 in its exponential form.

  Solution:

  \( \sqrt{55} \) can be written as \( 55^{1/2} \).

• Question 5: Add \( \sqrt{55} \) and \( \sqrt{20} \). Express your answer in simplest form.

  Solution:

  \( \sqrt{55} + \sqrt{20} \) remains \( \sqrt{55} + 2\sqrt{5} \), as neither term can be simplified further to combine them.

Use these practice questions to reinforce your understanding of simplifying and working with the square root of 55.

What is the square root of 55?

The square root of 55 is represented as \( \sqrt{55} \). It is approximately equal to 7.4162 when calculated using a calculator.

How do you write the square root of 55 in its simplified form?

The square root of 55 is already in its simplest radical form, \( \sqrt{55} \), as it cannot be simplified further.

Is the square root of 55 real?

Yes, the square root of 55 is a real number. It is a positive real number approximately equal to 7.4162.

Is \( \sqrt{55} \) a rational number?

No, \( \sqrt{55} \) is not a rational number. It is an irrational number because it cannot be expressed as a simple fraction of two integers.

How to find the square root of 55?

You can find the square root of 55 using a calculator by pressing the square root function key after entering 55. Alternatively, you can use methods such as the long division method or approximation techniques to find it manually.