Simplify Square Root of 108: A Step-by-Step Guide

The process of simplifying the square root of 108 involves expressing the number under the square root as a product of perfect squares and then simplifying. Here’s a detailed step-by-step method to simplify √108:

Steps to Simplify √108

1. Find the prime factorization of 108.
• 108 can be factored into 2 × 54
• 54 can be factored into 2 × 27
• 27 can be factored into 3 × 9
• 9 can be factored into 3 × 3

So, the prime factorization of 108 is:

\[108 = 2^2 \times 3^3\]

2. Group the prime factors into pairs.

We can rewrite 108 as:

\[108 = (2^2) \times (3^2) \times 3\]

3. Take the square root of each pair of factors.

We have:

\[\sqrt{108} = \sqrt{(2^2) \times (3^2) \times 3}\]

This simplifies to:

\[\sqrt{108} = 2 \times 3 \times \sqrt{3}\]

4. Multiply the results outside the square root.

The simplified form is:

\[2 \times 3 \times \sqrt{3} = 6\sqrt{3}\]

Final Simplified Form

Thus, the square root of 108 simplified is:

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

Verification

To verify, we can square the simplified form to see if we get back the original number:

\[(6\sqrt{3})^2 = 6^2 \times (\sqrt{3})^2 = 36 \times 3 = 108\]

This confirms that our simplification is correct.

Additional Information

Simplifying square roots is a fundamental skill in algebra that allows for easier manipulation and understanding of expressions involving radicals. The process involves breaking down a number into its prime factors and simplifying the radical by grouping pairs of prime factors. This not only makes the expression simpler but also helps in solving equations and comparing the sizes of numbers.

1. Prime Factorization:

   
   First, break down 108 into its prime factors. This is done by dividing 108 by the smallest prime number, 2, and continuing the process:

   • 108 ÷ 2 = 54
   • 54 ÷ 2 = 27
   • 27 ÷ 3 = 9
   • 9 ÷ 3 = 3
   • 3 ÷ 3 = 1

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

2. Group Factors:

   
   Group the prime factors into pairs:

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

3. Apply the Square Root:

   
   Take the square root of the grouped pairs:

   
   \( \sqrt{2^2} \times \sqrt{3^2 \times 3} = 2 \times 3 \times \sqrt{3} \)

4. Simplify:

   
   Simplify the expression:

   
   \( 2 \times 3 \times \sqrt{3} = 6\sqrt{3} \)

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

Understanding how to simplify square roots is essential for solving various mathematical problems, including those involving geometry, physics, and engineering. For instance, when calculating the length of the diagonal of a square room with an area of 108 square feet, simplifying the square root helps in determining the exact length quickly and efficiently.

Simplifying square roots like \( \sqrt{108} \) to \( 6\sqrt{3} \) not only aids in mathematical problem-solving but also enhances clarity and precision in mathematical expressions. By mastering this skill, students and professionals can tackle complex problems with greater ease and confidence.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25. The symbol for the square root is √.

Square roots can be simplified by factoring the number into its prime factors and grouping the factors into pairs. The square root of a product is the product of the square roots of the factors.

• Identify the perfect square factors of the number.
• Rewrite the number as a product of these factors.
• Simplify by taking the square root of the perfect square factors.

Simplifying square roots involves expressing the square root in its simplest radical form. For example, the square root of 108 can be simplified as follows:

1. List the factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.
2. Identify the perfect square factors: 1, 4, 9, 36.
3. Divide 108 by the largest perfect square: 108 ÷ 36 = 3.
4. Calculate the square root of 36: √36 = 6.
5. Combine the results: √108 = 6√3.

Thus, the simplified form of the square root of 108 is 6√3. This process helps in understanding the properties of square roots and their applications in various mathematical problems.

In conclusion, simplifying square roots is an essential skill in algebra and higher mathematics. It allows for easier manipulation and understanding of numbers, making complex calculations more manageable.

Prime factorization is a powerful technique for simplifying square roots. It involves breaking down the number into its prime factors and then simplifying the square root based on these factors. Here is a step-by-step guide to simplifying the square root of 108 using the prime factorization method:

1. Factorize the number 108 into its prime factors:
   • 108 is divisible by 2, giving 108 = 2 × 54
   • 54 is divisible by 2, giving 54 = 2 × 27
   • 27 is divisible by 3, giving 27 = 3 × 9
   • 9 is divisible by 3, giving 9 = 3 × 3
2. Write the prime factorization of 108:

   108 = 2 × 2 × 3 × 3 × 3

3. Group the prime factors into pairs:

   \(\sqrt{108} = \sqrt{2 \times 2 \times 3 \times 3 \times 3}\)

4. Take one number from each pair out of the square root:

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

   \(\sqrt{3 \times 3} = 3\)

5. Multiply these numbers together and place the remaining factor under the square root:

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

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

Verifying the simplified form of the square root of 108 involves checking that the simplified form, \(6\sqrt{3}\), is accurate. To do this, follow these steps:

1. Recall the simplified form:
   
   
   \(\sqrt{108} = 6\sqrt{3}\)

2. Square both sides to verify:
   
   
   \((6\sqrt{3})^2 = 108\)

3. Calculate the left side:
   
   
   \(6^2 \times (\sqrt{3})^2 = 36 \times 3 = 108\)

4. Compare the results:
   
   
   Since both sides equal 108, the simplified form \(6\sqrt{3}\) is verified to be correct.

This step-by-step verification confirms that \(6\sqrt{3}\) is indeed the simplest radical form of \(\sqrt{108}\).

To understand the square root of 108, we need to break it down using prime factorization and simplification methods. Let's explore this step by step.

First, consider the prime factorization of 108. We factorize it as follows:

• 108 is an even number, so it's divisible by 2: 108 = 2 × 54.
• Next, 54 is also even, so it's divisible by 2: 54 = 2 × 27.
• Finally, 27 can be factorized into: 27 = 3 × 9, and 9 = 3 × 3.

So, the prime factorization of 108 is: 108 = 2 × 2 × 3 × 3 × 3.

To simplify the square root of 108, we use the property of square roots which states that the square root of a product is the product of the square roots:

\[
\sqrt{108} = \sqrt{2 \times 2 \times 3 \times 3 \times 3} = \sqrt{2^2 \times 3^2 \times 3}
\]

We can pair the prime factors and take the square root of each pair:

\[
\sqrt{108} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{3} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3}
\]

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

In decimal form, the value of \(6\sqrt{3}\) can be approximated by multiplying 6 with the square root of 3, which is approximately 1.732:

\[
6 \times 1.732 \approx 10.392
\]

So, \(\sqrt{108} \approx 10.392\) in decimal form.

When simplifying the square root of 108, there are several common mistakes that students and learners might make. Understanding these mistakes can help avoid them and ensure accurate calculations.

• Incorrect Prime Factorization: One of the first steps in simplifying square roots is performing the prime factorization of the number. A common mistake is incorrectly identifying the prime factors of 108. The correct prime factorization of 108 is 2 × 2 × 3 × 3 × 3. Double-check the factors to ensure they are all prime numbers.
• Not Pairing Prime Factors Correctly: Another mistake is not correctly pairing the prime factors to bring them out of the square root. For example, for √108, you should pair the 2's and the 3's: √(2² × 3² × 3). Each pair of prime factors can be taken out of the square root as a single number.
• Forgetting to Multiply Factors Outside the Radical: After pairing the prime factors, it is crucial to multiply them correctly outside the radical. For instance, in √(2² × 3² × 3), the 2 and 3 that come out of the square root should be multiplied together: 2 × 3 = 6, giving 6√3. Forgetting this step can lead to incorrect simplification.
• Incorrectly Simplifying the Remaining Radical: After taking out the paired factors, some students forget to leave the remaining factor inside the radical. For √108, after taking out 6, we are left with √3 inside the radical, giving the final simplified form as 6√3. Always check if any factors remain inside the radical.
• Arithmetic Errors: Basic arithmetic mistakes, such as incorrect multiplication or addition, can lead to wrong results. Carefully perform each step and double-check calculations to avoid these errors.

By being aware of these common mistakes and carefully following each step, you can accurately simplify square roots, including √108.

The simplified form of the square root of 108, which is \(6\sqrt{3}\), has several practical applications in various fields of science, engineering, mathematics, and everyday life. Here are some notable applications:

• Geometry and Trigonometry:

  Simplified square roots are frequently used in geometry and trigonometry to simplify the computation of lengths, areas, and other properties of geometric figures. For example, the simplified form of square roots can make it easier to work with the Pythagorean theorem in right triangles.

• Engineering and Construction:

  Engineers and architects use simplified square roots to calculate distances, angles, and other important measurements in their designs. For instance, determining the diagonal length of a rectangular space often involves calculating the square root of sums of squares.

• Physics:

  In physics, simplified square roots appear in formulas for calculating energy, force, and motion. For example, they are used in equations involving kinetic energy, which is proportional to the square of the velocity.

• Computer Science:

  In computer algorithms, especially those related to graphics and simulations, simplified square roots help in optimizing calculations involving distances and magnitudes, leading to more efficient code execution.

• Finance:

  In finance, simplified square roots are used in statistical calculations, such as the standard deviation, which measures the amount of variation or dispersion of a set of values.

• Everyday Life:

  In daily activities, simplified square roots help in tasks like measuring areas for home improvement projects, understanding dimensions, and converting units. For example, when determining the size of a square area given its diagonal, the square root calculation becomes necessary.

Understanding how to simplify square roots and apply them in various contexts not only enhances problem-solving skills but also provides practical tools for a wide range of real-world situations.

• What is the square root of 108?

  The square root of 108 is \( \sqrt{108} \approx \pm10.392 \). The exact simplified form is \( 6\sqrt{3} \).

• How do you simplify the square root of 108?

  To simplify \( \sqrt{108} \), you can use the prime factorization method:

  1. Find the prime factors of 108: \( 108 = 2 \times 2 \times 3 \times 3 \times 3 \).
  2. Group the prime factors into pairs: \( \sqrt{2^2 \times 3^2 \times 3} \).
  3. Take the square root of each pair: \( 2 \times 3 \times \sqrt{3} \).
  4. Combine the results: \( 6\sqrt{3} \).

• Is the square root of 108 a rational number?

  No, \( \sqrt{108} \) is an irrational number because it cannot be expressed as a simple fraction. Its decimal representation is non-terminating and non-repeating.

• Can you show the long division method to find the square root of 108?

  Yes, the long division method provides a way to find the square root of 108 more accurately:

  1. Start with 108.00 and group the digits into pairs from the right: 1 08.00.
  2. Find the largest number whose square is less than or equal to the leftmost group (1). The number is 1, and the first digit of the quotient is 1.
  3. Subtract 1 from 1 to get 0, then bring down the next pair of digits (08).
  4. Double the quotient and use it as the new divisor (2). Find a number that, when added to 20 and multiplied by itself, gives a product less than or equal to 8. This number is 0.
  5. Continue the process with the remaining pairs of digits until you reach the desired level of accuracy. The quotient approximates \( \sqrt{108} \approx 10.392 \).

• How do you use the average method to approximate the square root of 108?

  The average method involves finding two perfect squares closest to 108 and averaging their square roots. For example, \( \sqrt{100} = 10 \) and \( \sqrt{121} = 11 \). Averaging them gives \( (10 + 11) / 2 = 10.5 \), which is close to \( \sqrt{108} \).

• What is the use of the square root of 108 in real life?

  Square roots are used in various fields such as architecture, engineering, and physics. For instance, if a room has an area of 108 square feet, the length of each side of the room (assuming it is square) can be found by calculating \( \sqrt{108} \approx 10.39 \) feet.

Simplifying the square root of 108 is a valuable skill that enhances your understanding of radical expressions and their applications. The process of simplifying \( \sqrt{108} \) to \( 6\sqrt{3} \) involves recognizing perfect square factors and using them to break down the expression into a simpler form.

By following the steps of prime factorization and grouping factors into pairs, you can systematically simplify any square root expression. This method not only makes complex calculations more manageable but also helps in various real-life applications, such as geometry and algebra.

Understanding the simplification process also aids in recognizing and avoiding common mistakes, such as incorrect factorization or misapplication of the product rule for radicals. Being thorough with the fundamentals ensures accuracy and efficiency in solving mathematical problems.

Moreover, the ability to simplify square roots is useful in many fields. For instance, it is crucial in solving quadratic equations, optimizing functions, and even in physics for simplifying wave functions or distances.

Overall, mastering the simplification of square roots, like \( \sqrt{108} \), strengthens your mathematical foundation and equips you with tools to tackle more advanced topics with confidence.