Square Root of 108 Simplified Radical Form: Master the Simplification Process

The square root of 108 can be simplified using the properties of radicals and prime factorization. Below is the detailed step-by-step process to simplify the square root of 108:

Step-by-Step Simplification

1. Start with the number 108.
2. Find the prime factors of 108:
   • 108 ÷ 2 = 54
   • 54 ÷ 2 = 27
   • 27 ÷ 3 = 9
   • 9 ÷ 3 = 3
   • 3 ÷ 3 = 1
3. So, the prime factorization of 108 is: 108 = 2 × 2 × 3 × 3 × 3.
4. Group the prime factors into pairs: (2 × 2) and (3 × 3) and the remaining 3.
5. Rewrite the square root of 108 using these pairs:

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

6. Simplify by taking the square root of each pair of prime factors:

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

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

7. Combine these simplified terms outside the radical and leave the remaining factor inside the radical:

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

8. Multiply the numbers outside the radical:

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

Conclusion

The simplified radical form of the square root of 108 is:

Simplified radical form is a way of expressing square roots in their simplest form. It involves breaking down a number into its prime factors and simplifying the expression by extracting square roots of any perfect square factors. This makes the expression easier to work with and understand.

To understand the simplified radical form, let's start with the concept of a square root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 × 3 = 9. When a number is not a perfect square, its square root can often be simplified using radical notation.

The radical sign (√) is used to denote the square root of a number. Simplifying a square root means finding the most compact form of this expression. For instance, the square root of 108 can be simplified to a simpler radical form.

Here is a step-by-step guide to simplifying the square root of 108:

1. Prime Factorization: Break down 108 into its prime factors. 108 = 2 × 54 = 2 × 2 × 27 = 2 × 2 × 3 × 9 = 2 × 2 × 3 × 3 × 3.
2. Group the Factors: Pair up the prime factors. In this case, we have (2 × 2) and (3 × 3) and a single 3 left over.
3. Extract the Square Roots: Take the square root of each pair. The square root of 2 × 2 is 2, and the square root of 3 × 3 is 3.
4. Combine the Results: Multiply the results of the square roots together along with any remaining factors inside the radical. Here, 2 × 3 = 6, and we are left with √3.
5. Final Simplified Form: The square root of 108 in its simplest radical form is 6√3.

By following these steps, we can see that the square root of 108 simplifies to 6√3, making it easier to work with in various mathematical problems and applications.

Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. The symbol for the square root is √, called the radical sign, and the number inside this symbol is known as the radicand.

To understand square roots, it is essential to recognize perfect squares. A perfect square is a number that has an integer as its square root. For example, the numbers 1, 4, 9, 16, and 25 are perfect squares because their square roots are 1, 2, 3, 4, and 5, respectively.

However, not all numbers are perfect squares. When dealing with non-perfect squares, the square root is often an irrational number, meaning it cannot be expressed as a simple fraction and its decimal form is non-terminating and non-repeating. For instance, the square root of 108 is not a perfect square and is an irrational number.

To simplify a square root, such as √108, we use the prime factorization method. This involves breaking down the radicand into its prime factors and pairing identical factors. Let's illustrate this process step-by-step:

1. Prime factorize the number 108: 108 = 2 × 2 × 3 × 3 × 3.
2. Group the prime factors into pairs: (2 × 2) and (3 × 3) and 3.
3. For each pair, take one factor out of the radical: √(2 × 2) × √(3 × 3) × √3 = 2 × 3 × √3.
4. Multiply the numbers outside the radical: 2 × 3 = 6.
5. Combine the result with the remaining factor inside the radical: 6√3.

Thus, the simplified radical form of √108 is 6√3.

Understanding and simplifying square roots is crucial in various mathematical applications, from solving quadratic equations to calculating distances in geometry. Mastering these skills can enhance your problem-solving abilities and mathematical proficiency.

The prime factorization method is a systematic approach to simplify the square root of a number by expressing it as a product of its prime factors. Let's apply this method to simplify the square root of 108.

1. Find the Prime Factors:

   First, we need to determine the prime factors of 108. The prime factorization of 108 is:

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

2. Group the Prime Factors into Pairs:

   Next, we pair the prime factors. Since we are looking for the square root, we group the prime factors into pairs:

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

3. Simplify by Taking the Square Root of Each Pair:

   Now, take the square root of each pair of prime factors:

   \[
   \sqrt{2^2} = 2 \quad \text{and} \quad \sqrt{3^2} = 3
   \]

   Since we have an unpaired 3, it remains under the radical:

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

4. Combine the Simplified Terms:

   The final simplified form of the square root of 108 is:

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

This method not only simplifies the square root but also helps in understanding the structure of the number in terms of its prime factors.

To simplify the square root of 108, we will follow a systematic process involving the prime factorization method. Here are the detailed steps:

1. Prime Factorization:

   First, find the prime factors of 108. The prime factorization of 108 is:

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

2. Group the Prime Factors:

   Next, group the prime factors into pairs of equal factors:

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

3. Extract the Square Roots:

   Take the square root of each pair and simplify:

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

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

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

4. Combine the Results:

   Multiply the extracted square roots:

   \[ 2 \times 3 = 6 \]

   Thus, the simplified form is:

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

Following these steps ensures that you simplify the square root of 108 accurately to its simplest radical form, which is \(6\sqrt{3}\).

To simplify the square root of 108, we need to use the prime factorization method. This involves breaking down the number 108 into its prime factors and then pairing these factors to simplify the square root.

1. Prime Factorization of 108:

   First, we find the prime factors of 108. We can do this by repeatedly dividing by the smallest prime numbers until we reach 1.

   • 108 is divisible by 2: 108 ÷ 2 = 54
   • 54 is divisible by 2: 54 ÷ 2 = 27
   • 27 is divisible by 3: 27 ÷ 3 = 9
   • 9 is divisible by 3: 9 ÷ 3 = 3
   • 3 is divisible by 3: 3 ÷ 3 = 1

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

2. Pairing the Prime Factors:

   Next, we pair the prime factors. A pair of the same factors can be taken out of the square root as a single factor.

   • Pair of 2s: \( \sqrt{2 \times 2} = 2 \)
   • Pair of 3s: \( \sqrt{3 \times 3} = 3 \)
   • Remaining factor: \( \sqrt{3} \)
3. Combining the Pairs:

   We now combine the simplified factors outside the square root with any remaining factors inside the square root:

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

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

To simplify the square root of 108 using the method of extracting square roots from pairs, follow these detailed steps:

1. Prime Factorization: Start by finding the prime factors of 108. The prime factorization of 108 is:
   • 108 = 2 × 54
   • 54 = 2 × 27
   • 27 = 3 × 9
   • 9 = 3 × 3
   So, 108 = 2 × 2 × 3 × 3 × 3.
2. Group the Factors into Pairs: Group the prime factors into pairs of equal numbers:
   • (2 × 2) and (3 × 3) and 3.
3. Extract the Square Roots of the Pairs: Take the square root of each pair:
   • √(2 × 2) = 2
   • √(3 × 3) = 3
4. Multiply the Results: Multiply the results of the square roots extracted from the pairs:
   • 2 × 3 = 6
5. Include the Remaining Factor: Multiply by the remaining factor under the square root:
   • 6 × √3 = 6√3
6. Final Simplified Form: Therefore, the square root of 108 in its simplest radical form is:
   • √108 = 6√3

This method ensures that the square root of 108 is simplified accurately by systematically pairing and extracting the square roots of prime factors.

Once we have extracted the square roots from pairs, the next step is to combine the simplified terms to get the final simplified radical form. Let's walk through the process step-by-step:

1. Recall that we have already simplified the square root of 108 to \(6\sqrt{3}\). This was done by pairing the prime factors and extracting the square roots from those pairs.
2. We factorized 108 as follows:
   • 108 = 2 × 54
   • 54 = 2 × 27
   • 27 = 3 × 9
   • 9 = 3 × 3
   • So, 108 = 2 × 2 × 3 × 3 × 3
3. We paired the prime factors:
   • Pair of 2s: \(2^2\)
   • Pair of 3s: \(3^2\)
   • Remaining 3
4. We extracted the square roots from the pairs:
   • \(\sqrt{2^2} = 2\)
   • \(\sqrt{3^2} = 3\)
5. Combining these extracted terms with the remaining square root, we get:
   • \(2 \times 3 \times \sqrt{3} = 6\sqrt{3}\)

Therefore, the final simplified form of the square root of 108 is \(6\sqrt{3}\). This process of combining the simplified terms ensures that we have expressed the square root in its simplest radical form.

Understanding how to combine simplified terms effectively is crucial for solving similar problems and simplifying other radicals. Practice this method with different numbers to become more comfortable with the steps involved.

The final simplified form of the square root of 108 is derived by combining the steps of prime factorization, pairing prime factors, and extracting square roots from these pairs.

1. Prime Factorization: First, we decompose 108 into its prime factors:

   \[108 = 2 \times 54 = 2 \times 2 \times 27 = 2 \times 2 \times 3 \times 9 = 2^2 \times 3^3\]

2. Grouping Prime Factors: Next, we group the prime factors into pairs:

   \[2^2 \text{ and } 3^2 \text{ are perfect squares}\]

3. Extracting Square Roots: We take the square root of each pair:

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

4. Combining Simplified Terms: Finally, we combine the simplified terms to get the final form:

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

Therefore, the square root of 108 in its simplest radical form is:

\[6\sqrt{3}\]

This simplified form is more manageable and often easier to use in further calculations or applications.

Visualizing the simplification of the square root of 108 helps in understanding the process more clearly. Here are the steps represented visually:

Step 1: Prime Factorization

First, we find the prime factors of 108.

\[ 108 = 2 \times 54 = 2 \times 2 \times 27 = 2 \times 2 \times 3 \times 9 = 2 \times 2 \times 3 \times 3 \times 3 \]

This can be visualized as:

• 108
• 2 × 54
• 2 × 2 × 27
• 2 × 2 × 3 × 9
• 2 × 2 × 3 × 3 × 3

Step 2: Pairing the Prime Factors

Next, we pair the identical factors.

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

Pairing the factors, we get:

• \[ \sqrt{(2 \times 2) \times (3 \times 3) \times 3} \]

Step 3: Extracting the Pairs

We take the square root of each pair and bring them out of the radical sign.

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

Here is a visual representation:

• From \[ 2 \times 2 \], extract 2
• From \[ 3 \times 3 \], extract 3
• Multiply the extracted numbers: \[ 2 \times 3 = 6 \]
• Remain with \[ \sqrt{3} \]

Step 4: Final Simplified Form

Combining all the extracted factors, we have:

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

Example Problems

Here are a couple of examples to practice:

1. Find the simplified form of \[ \sqrt{72} \]

Solution: \[ \sqrt{72} = \sqrt{2^3 \times 3^2} = 6\sqrt{2} \]

2. Find the simplified form of \[ \sqrt{200} \]

Solution: \[ \sqrt{200} = \sqrt{2^3 \times 5^2} = 10\sqrt{2} \]

Visualizing these steps ensures a clear understanding of simplifying square roots and reinforces the concept with examples.

When simplifying the square root of 108, it's crucial to be aware of common pitfalls. Here are some mistakes to avoid to ensure accurate and simplified results:

• Incorrect Prime Factorization:

  Ensure that the number 108 is factorized correctly into its prime components. The correct factorization is \(108 = 2^2 \times 3^3\). Missing a factor or misidentifying a non-prime number can lead to errors in simplification.

• Forgetting to Pair Factors:

  When simplifying, remember to pair the prime factors. For \(108\), you should pair \(2 \times 2\) and \(3 \times 3\). This leads to \(\sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3}\).

• Not Simplifying Fully:

  Sometimes, intermediate steps are correct, but the final expression is not fully simplified. Always ensure that the final form is in its simplest radical form, like \(6\sqrt{3}\) for \(\sqrt{108}\).

• Incorrect Arithmetic:

  Errors in basic arithmetic during factorization or simplification can lead to wrong results. Double-check each step, especially when multiplying or dividing numbers.

• Misunderstanding the Radical Sign:

  Remember that the radical sign \(\sqrt{}\) applies to the entire product under it. Misinterpreting this can lead to incorrect simplification. For example, \(\sqrt{36 \times 3}\) should be correctly simplified to \(6\sqrt{3}\), not \(\sqrt{36} \times 3\).

• Ignoring Perfect Squares:

  Identify and use the largest perfect square factor for easier simplification. For \(108\), the largest perfect square factor is \(36\), which simplifies the process considerably.

By keeping these common mistakes in mind and carefully following the steps, you can accurately simplify the square root of 108 and other similar expressions.

Practicing the simplification of square roots helps reinforce the concepts and techniques needed to master the process. Here are some practice problems along with their solutions to help you understand how to simplify the square root of 108 and similar problems.

Practice Problems

1. Simplify \( \sqrt{108} \).
2. Simplify \( \sqrt{72} \).
3. Simplify \( \sqrt{200} \).
4. Simplify \( \sqrt{50} \).
5. Simplify \( \sqrt{147} \).

Solutions

1. \( \sqrt{108} \)

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

2. \( \sqrt{72} \)

   1. Prime factorize 72: \( 72 = 2^3 \times 3^2 \).
   2. Group the prime factors into pairs: \( 72 = (2^2 \times 2) \times (3^2) \).
   3. Take the square root of each pair: \( \sqrt{72} = \sqrt{2^2 \times 3^2 \times 2} = 2 \times 3 \times \sqrt{2} \).
   4. Simplified form: \( \sqrt{72} = 6\sqrt{2} \).

3. \( \sqrt{200} \)

   1. Prime factorize 200: \( 200 = 2^3 \times 5^2 \).
   2. Group the prime factors into pairs: \( 200 = (2^2 \times 2) \times (5^2) \).
   3. Take the square root of each pair: \( \sqrt{200} = \sqrt{2^2 \times 5^2 \times 2} = 2 \times 5 \times \sqrt{2} \).
   4. Simplified form: \( \sqrt{200} = 10\sqrt{2} \).

4. \( \sqrt{50} \)

   1. Prime factorize 50: \( 50 = 2 \times 5^2 \).
   2. Group the prime factors into pairs: \( 50 = 2 \times (5^2) \).
   3. Take the square root of each pair: \( \sqrt{50} = \sqrt{2 \times 5^2} = 5 \times \sqrt{2} \).
   4. Simplified form: \( \sqrt{50} = 5\sqrt{2} \).

5. \( \sqrt{147} \)

   1. Prime factorize 147: \( 147 = 3 \times 7^2 \).
   2. Group the prime factors into pairs: \( 147 = 3 \times (7^2) \).
   3. Take the square root of each pair: \( \sqrt{147} = \sqrt{3 \times 7^2} = 7 \times \sqrt{3} \).
   4. Simplified form: \( \sqrt{147} = 7\sqrt{3} \).

By practicing these problems, you can become proficient in simplifying square roots and better understand the underlying principles.

Simplified radicals play an important role in various real-life applications across multiple fields. Here are some key examples:

• Architecture and Construction: Architects and builders use radical expressions to calculate dimensions, areas, and volumes of different shapes. For instance, determining the diagonal of a square floor plan requires the square root of the sum of the squares of the sides.
• Engineering: In electrical engineering, the square root function is crucial for calculating electrical properties such as impedance and power. Simplified radicals help engineers design circuits and systems more efficiently.
• Physics: Radical expressions are used in formulas to describe natural phenomena. For example, the formula for the period of a pendulum involves the square root of its length divided by the acceleration due to gravity.
• Biology: Biologists use radical expressions to compare surface areas and volumes of different organisms. This is particularly useful in understanding metabolic rates and other physiological properties.
• Finance: In the financial industry, simplified radicals are used in formulas to calculate interest rates, depreciation, and inflation. For example, the compound interest formula involves the square root to determine the annual growth rate.

Here is a detailed example of how simplified radicals are used in a real-life scenario:

Example: Calculating the Diagonal of a Square

Suppose you have a square with each side measuring 10 meters, and you want to find the length of the diagonal.

1. Use the Pythagorean theorem for a right triangle: \( a^2 + b^2 = c^2 \).
2. Since both sides of the square are equal, the equation becomes: \( 10^2 + 10^2 = c^2 \).
3. Simplify the equation: \( 100 + 100 = c^2 \), so \( 200 = c^2 \).
4. Take the square root of both sides to find the diagonal: \( c = \sqrt{200} \).
5. Simplify the radical: \( \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \).

Therefore, the length of the diagonal is \( 10\sqrt{2} \) meters, which is an application of simplified radicals in determining precise measurements in construction.

1. What is the square root of 108?

The square root of 108 is \(\sqrt{108} = \pm 10.392\). The simplified radical form is \(6\sqrt{3}\).

2. How is the square root of 108 simplified?

The square root of 108 is simplified by finding its prime factorization and pairing the factors:

• Prime factorization of 108: \(108 = 2 \times 2 \times 3 \times 3 \times 3\)
• Pair the factors: \(\sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3}\)

3. Is the square root of 108 a rational number?

No, the square root of 108 is not a rational number because it cannot be expressed as a simple fraction. The value of \(\sqrt{3}\) is an irrational number, making \(6\sqrt{3}\) irrational as well.

4. What is the decimal value of the square root of 108?

The decimal value of \(\sqrt{108}\) is approximately 10.392 when rounded to three decimal places.

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

The long division method involves the following steps:

1. Write 108.00 and pair the digits from the right.
2. Find a number that, when squared, is less than or equal to 1 (the first pair).
3. Double the quotient for the next divisor and continue the process, bringing down pairs of zeros and finding suitable digits.
4. Repeat until the desired accuracy is achieved.

6. Why is 108 not a perfect square?

108 is not a perfect square because its square root is not an integer. The prime factorization includes unpaired factors (3), resulting in an irrational number.

7. How do you use the Babylonian method to find the square root of 108?

The Babylonian method involves iterative averaging:

1. Start with an initial guess (e.g., 54).
2. Divide 108 by the guess and average the result with the guess.
3. Repeat the process until the difference between successive guesses is less than a chosen threshold (e.g., 0.001).
4. After several iterations, you will find \(\sqrt{108} \approx 10.392\).

The process of simplifying the square root of 108 involves breaking down the number into its prime factors and then extracting the square root of those factors. This results in the simplest radical form.

To summarize:

• The square root of 108 is an irrational number, meaning it cannot be expressed as a simple fraction and has a non-terminating, non-repeating decimal expansion.
• The simplified radical form of the square root of 108 is \( 6\sqrt{3} \). This is obtained by recognizing that 108 can be factored into \( 2^2 \times 3^3 \), and then taking the square root of the pairs of factors.
• Mathematically, the simplification can be expressed as: \[ \sqrt{108} = \sqrt{2^2 \times 3^3} = \sqrt{2^2} \times \sqrt{3^3} = 2 \times 3 \sqrt{3} = 6\sqrt{3} \]
• The approximate decimal value of \( \sqrt{108} \) is 10.3923, which is useful for practical applications but less precise than the exact simplified radical form.

Understanding the concept of simplifying radicals is essential in various mathematical applications, from solving equations to real-world problem-solving where exact values are necessary. The process demonstrated here can be applied to other numbers to find their simplest radical forms.

In conclusion, mastering the simplification of radicals like \( \sqrt{108} \) enhances mathematical fluency and provides a foundation for more advanced mathematical studies.