Square Root 108 Simplified: Easy Steps to Understand and Apply

The square root of 108 simplified is expressed as follows:

\[ \sqrt{108} = 6\sqrt{3} \]

The square root of a number is a value that, when multiplied by itself, gives the original number. It is one of the fundamental concepts in mathematics, often represented by the radical symbol \( \sqrt{} \). For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

Square roots are essential for solving quadratic equations, understanding geometric properties, and in various real-world applications such as physics and engineering.

Below is a step-by-step guide to understanding square roots:

1. Identify the number: Determine the number for which you need to find the square root.
2. Prime Factorization: Break down the number into its prime factors. This helps in simplifying square roots.
3. Simplify: Pair the prime factors and move them outside the radical.

For example, to find the square root of 108:

• Prime factorize 108: \( 108 = 2 \times 2 \times 3 \times 3 \times 3 \).
• Group the factors in pairs: \( 108 = (2 \times 2) \times (3 \times 3) \times 3 \).
• Simplify by taking one number from each pair: \( \sqrt{108} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3} \).

Thus, the simplified form of \( \sqrt{108} \) is \( 6\sqrt{3} \).

The square root of 108 is a value that, when multiplied by itself, results in 108. This can be expressed as \( \sqrt{108} \). To better understand this, we need to look into its simplification process.

The process of simplifying the square root of 108 involves breaking it down into its prime factors:

1. Prime Factorization: Begin by finding the prime factors of 108.
   • 108 can be divided by 2: \( 108 \div 2 = 54 \)
   • 54 can be divided by 2: \( 54 \div 2 = 27 \)
   • 27 can be divided by 3: \( 27 \div 3 = 9 \)
   • 9 can be divided by 3: \( 9 \div 3 = 3 \)
   • 3 is a prime number.
2. Write Prime Factorization: The prime factorization of 108 is \( 2 \times 2 \times 3 \times 3 \times 3 \).
3. Group the Factors: Group the factors into pairs of the same number.
   • \( (2 \times 2) \) and \( (3 \times 3) \), with one 3 left unpaired.
4. Simplify the Radical: For each pair of factors, take one factor out of the radical.
   • \( \sqrt{2 \times 2 \times 3 \times 3 \times 3} = \sqrt{(2^2) \times (3^2) \times 3} \)
   • Take out one 2 and one 3: \( 2 \times 3 \sqrt{3} = 6\sqrt{3} \)

Thus, the simplified form of \( \sqrt{108} \) is \( 6\sqrt{3} \). This means \( 6\sqrt{3} \times 6\sqrt{3} = 108 \).

Understanding the square root of 108 and its simplification can help in various mathematical problems and in gaining a deeper insight into the properties of numbers.

Simplifying the square root of 108 involves breaking down the number into its prime factors and then simplifying the radical expression. Here is a detailed, step-by-step process to achieve this:

1. Find the Prime Factors of 108:
   • Start by dividing 108 by the smallest prime number, which is 2: \( 108 \div 2 = 54 \).
   • Divide 54 by 2 again: \( 54 \div 2 = 27 \).
   • Next, divide 27 by 3: \( 27 \div 3 = 9 \).
   • Finally, divide 9 by 3: \( 9 \div 3 = 3 \).
   • 3 is a prime number and cannot be divided further.

   So, the prime factorization of 108 is \( 2 \times 2 \times 3 \times 3 \times 3 \).

2. Group the Prime Factors:

   Pair the prime factors to simplify under the radical:

   • \( 108 = 2 \times 2 \times 3 \times 3 \times 3 = (2 \times 2) \times (3 \times 3) \times 3 \).
3. Simplify the Radical Expression:

   Take one number from each pair out of the radical:

   • \( \sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} = \sqrt{2^2 \times 3^2 \times 3} \).
   • Simplify by taking out the squares: \( 2 \times 3 \sqrt{3} = 6\sqrt{3} \).

Therefore, the simplified form of \( \sqrt{108} \) is \( 6\sqrt{3} \). This process not only helps in understanding the simplification of square roots but also reinforces the concept of prime factorization.

The prime factorization method is a systematic way of breaking down a number into its prime factors, which can then be used to simplify square roots. Here is a step-by-step guide to applying the prime factorization method to simplify the square root of 108:

1. Identify the Number:

   We start with the number 108.

2. Divide by the Smallest Prime Number:

   Begin by dividing 108 by the smallest prime number, which is 2:

   • \( 108 \div 2 = 54 \)
3. Continue Dividing by Prime Numbers:

   Continue the process with the quotient until all factors are prime:

   • \( 54 \div 2 = 27 \)
   • \( 27 \div 3 = 9 \)
   • \( 9 \div 3 = 3 \)
   • \( 3 \div 3 = 1 \)
4. List All Prime Factors:

   Collect all the prime factors identified through division:

   • The prime factorization of 108 is \( 2 \times 2 \times 3 \times 3 \times 3 \).
5. Group the Prime Factors:

   Group the prime factors in pairs to simplify the square root:

   • \( 108 = (2 \times 2) \times (3 \times 3) \times 3 \)
6. Simplify the Radical:

   Take one factor from each pair outside the radical:

   • \( \sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} \)
   • \( \sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} \)
   • \( \sqrt{108} = 2 \times 3 \times \sqrt{3} \)
   • \( \sqrt{108} = 6\sqrt{3} \)

By using the prime factorization method, we have simplified \( \sqrt{108} \) to \( 6\sqrt{3} \). This method not only helps in simplifying square roots but also strengthens the understanding of prime numbers and their properties.

The factor tree method is a visual way of breaking down a number into its prime factors, which can then be used to simplify square roots. Here is a detailed, step-by-step guide to using the factor tree to simplify the square root of 108:

1. Start with the Number:

   Begin with the number 108 at the top of the tree.

2. Divide by the Smallest Prime Number:

   Divide 108 by the smallest prime number, which is 2:

   • 108 divides into 2 and 54, so write 2 and 54 as branches.

   
   

   
   
   
   
   
   
   
   
   

   108
   2	54

   
   


3. Continue the Process:

   Divide each branch by the smallest prime number:

   • 54 divides into 2 and 27, so write 2 and 27 as branches under 54.
   • 27 divides into 3 and 9, so write 3 and 9 as branches under 27.
   • 9 divides into 3 and 3, so write 3 and 3 as branches under 9.

   
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   108
   2	54
   	2	27
   		3	9
   			3	3

   
   


4. Identify the Prime Factors:

   Once you reach the end of each branch, you will have the prime factors of 108:

   • The prime factors are \( 2, 2, 3, 3, 3 \).
5. Group the Prime Factors:

   Group the prime factors in pairs to simplify the square root:

   • \( 108 = (2 \times 2) \times (3 \times 3) \times 3 \).
6. Simplify the Radical:

   Take one factor from each pair outside the radical:

   • \( \sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} \)
   • \( \sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} \)
   • \( \sqrt{108} = 2 \times 3 \times \sqrt{3} \)
   • \( \sqrt{108} = 6\sqrt{3} \)

Using the factor tree method, we have successfully simplified \( \sqrt{108} \) to \( 6\sqrt{3} \). This method provides a clear visual representation of the prime factorization process, making it easier to understand and apply.

Simplifying the square root of 108 involves breaking down the number into its prime factors and then simplifying the radical expression. Here are the detailed steps to simplify \( \sqrt{108} \):

1. Find the Prime Factors of 108:

   Begin by finding the prime factors of 108.

   • 108 can be divided by 2: \( 108 \div 2 = 54 \).
   • 54 can be divided by 2 again: \( 54 \div 2 = 27 \).
   • 27 can be divided by 3: \( 27 \div 3 = 9 \).
   • 9 can be divided by 3: \( 9 \div 3 = 3 \).
   • 3 is a prime number.
2. Write Prime Factorization:

   Write down the prime factors of 108:

   • The prime factorization of 108 is \( 2 \times 2 \times 3 \times 3 \times 3 \).
3. Group the Prime Factors:

   Group the prime factors in pairs to simplify under the radical:

   • \( 108 = (2 \times 2) \times (3 \times 3) \times 3 \).
4. Simplify the Radical Expression:

   Take one factor from each pair outside the radical:

   • \( \sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} \).
   • \( \sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} \).
   • \( \sqrt{108} = 2 \times 3 \times \sqrt{3} \).
   • \( \sqrt{108} = 6\sqrt{3} \).

By following these steps, we have simplified \( \sqrt{108} \) to \( 6\sqrt{3} \). This method not only helps in understanding the simplification of square roots but also reinforces the concept of prime factorization.

Expressing \( \sqrt{108} \) in radical form involves simplifying the expression by using prime factorization. Here are the detailed steps to express \( \sqrt{108} \) in its simplest radical form:

1. Find the Prime Factors of 108:

   Start by determining the prime factors of 108:

   • 108 can be divided by 2: \( 108 \div 2 = 54 \)
   • 54 can be divided by 2 again: \( 54 \div 2 = 27 \)
   • 27 can be divided by 3: \( 27 \div 3 = 9 \)
   • 9 can be divided by 3: \( 9 \div 3 = 3 \)
   • 3 is a prime number.
2. Write Down the Prime Factorization:

   Record the prime factorization of 108:

   • The prime factorization of 108 is \( 2 \times 2 \times 3 \times 3 \times 3 \).
3. Group the Prime Factors:

   Group the prime factors in pairs to simplify under the radical:

   • \( 108 = (2 \times 2) \times (3 \times 3) \times 3 \).
4. Simplify the Radical Expression:

   Simplify the expression by taking one factor from each pair outside the radical:

   • \( \sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} \)
   • \( \sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} \)
   • \( \sqrt{108} = 2 \times 3 \times \sqrt{3} \)
   • \( \sqrt{108} = 6\sqrt{3} \)

Thus, the simplified radical form of \( \sqrt{108} \) is \( 6\sqrt{3} \). This method demonstrates how prime factorization can be used to simplify square roots effectively, making complex expressions easier to work with in various mathematical contexts.

Converting \( \sqrt{108} \) to its decimal form involves a simple calculation that can be done using a calculator. Here are the detailed steps to convert \( \sqrt{108} \) to a decimal:

1. Understand the Simplified Radical Form:

   We have already simplified \( \sqrt{108} \) to \( 6\sqrt{3} \).

2. Use a Calculator to Find the Decimal Value of \( \sqrt{3} \):

   Find the decimal value of \( \sqrt{3} \) using a calculator:

   • \( \sqrt{3} \approx 1.732 \)
3. Multiply the Simplified Value by the Decimal:

   Multiply the simplified radical form by the decimal value obtained:

   • \( 6 \times 1.732 = 10.392 \)

Therefore, the decimal form of \( \sqrt{108} \) is approximately 10.392. This conversion is useful in practical applications where a decimal representation is needed for calculations.

The simplified form of square roots is widely used in various fields and everyday situations. Understanding how to simplify square roots can provide numerous benefits, especially in mathematics, science, engineering, and even daily life. Here are some common applications:

• Mathematical Calculations: Simplified square roots make complex calculations easier to handle. For example, the square root of 108 simplified is 6√3, which can simplify addition, subtraction, multiplication, and division involving roots.
• Geometry: In geometry, square roots are often used to calculate lengths of sides in right triangles (via the Pythagorean theorem) and other shapes. Knowing the simplified form helps in quick and accurate measurements.
• Physics: Simplified square roots are used in various physics equations, such as calculating energy, force, and wave frequencies. This simplification aids in solving problems more efficiently.
• Engineering: Engineers use square roots in designing structures, electronics, and various mechanical systems. Simplified forms make it easier to work with complex formulas and ensure precision.
• Computer Science: Algorithms often involve square roots. Simplifying these roots can enhance algorithm efficiency and performance in computational tasks.
• Statistics: In statistics, square roots are used in formulas for standard deviation and variance. Simplifying these roots helps in clearer and more understandable data analysis.
• Daily Life: Simplified square roots can be useful in everyday situations, such as calculating areas, understanding proportions, and working with measurements in cooking or home improvement projects.

By mastering the simplification of square roots, you can enhance your problem-solving skills and apply them effectively across a wide range of disciplines and real-world scenarios.

Here are some common questions about the square root of 108:

• What is the square root of 108?

  The square root of 108 is \( \sqrt{108} \). In its simplest radical form, it is expressed as \( 6\sqrt{3} \).

• Is 108 a perfect square?

  No, 108 is not a perfect square because its square root \( \sqrt{108} \) is not an integer. The exact value of \( \sqrt{108} \) is approximately 10.392.

• Is the square root of 108 a rational number?

  No, the square root of 108 is an irrational number because it cannot be expressed as a simple fraction and its decimal form is non-terminating and non-repeating.

• How can you simplify the square root of 108?

  To simplify \( \sqrt{108} \), you can factorize 108 into 36 and 3, where 36 is a perfect square. Thus, \( \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} \).

• What is the decimal form of \( \sqrt{108} \)?

  The decimal form of \( \sqrt{108} \) is approximately 10.392, correct to three decimal places.

• How do you find the square root of 108 using the long division method?

  The long division method can be used to find more precise values of \( \sqrt{108} \) step-by-step. You pair the digits of 108 from right to left and perform division, similar to traditional long division, to achieve a result accurate to multiple decimal places.

In conclusion, the square root of 108, simplified as \(6\sqrt{3}\), provides a clear example of how to handle non-perfect squares in mathematics. By understanding the process of simplification, we can appreciate the beauty of breaking down complex numbers into simpler, more manageable forms.

The journey to simplifying \(\sqrt{108}\) involved recognizing that 108 can be factored into \(36 \times 3\), where 36 is a perfect square. This allowed us to express \(\sqrt{108}\) as \(\sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}\). This form not only makes calculations easier but also helps in various practical applications such as geometry and algebra.

Understanding the square root of 108 also highlights the importance of recognizing irrational numbers. While the simplified form \(6\sqrt{3}\) is useful, its decimal approximation, approximately 10.392, is often used in practical applications where a numerical value is needed.

The methods we explored, from prime factorization to using the factor tree, are essential tools in the mathematical toolkit. They demonstrate that even complex problems can be broken down into simpler steps, making them easier to solve.

Ultimately, the square root of 108 is a small but significant example of how mathematics allows us to understand and manipulate numbers in powerful ways. Whether in academic pursuits or real-world applications, the ability to simplify square roots is a fundamental skill that enhances our problem-solving capabilities.