108 Square Root Simplified: Master the Simplification Process Easily

The square root of 108 can be simplified by factoring out perfect squares from the number 108. Below is a step-by-step process to simplify the square root of 108:

Step-by-Step Simplification

1. Identify the prime factorization 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.

   Therefore, the prime factorization of 108 is: \(108 = 2^2 \times 3^3\).

2. Group the prime factors into pairs:
   • \(2^2\) is a pair.
   • \(3^2\) is a pair, with one 3 left over.
3. Rewrite the square root of 108 using these pairs: \[ \sqrt{108} = \sqrt{2^2 \times 3^2 \times 3} \]
4. Take the square root of the pairs and multiply by the remaining factor:
   • \(\sqrt{2^2} = 2\)
   • \(\sqrt{3^2} = 3\)
   • The remaining factor is 3.
   \[ \sqrt{108} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3} \]

Conclusion

Therefore, the simplified form of the square root of 108 is:

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16. The symbol for the square root is √. For a number x, the square root is represented as √x.

Understanding square roots is essential for various mathematical operations and applications. They are fundamental in solving quadratic equations, understanding geometric properties, and simplifying expressions in algebra and calculus.

Here are some key points about square roots:

• Perfect Squares: These are numbers whose square roots are whole numbers. Examples include 1, 4, 9, 16, 25, and so on. For instance, √25 = 5.
• Non-Perfect Squares: These are numbers whose square roots are not whole numbers and are often irrational. For example, √2 and √3 are irrational numbers.
• Principal Square Root: The non-negative square root of a number. For example, the principal square root of 9 is 3, not -3.

When dealing with square roots, especially those of non-perfect squares, it's often useful to simplify them to make calculations easier. This process involves breaking down the number inside the square root into its prime factors and identifying any perfect squares within those factors.

In the sections that follow, we will explore the steps to simplify the square root of 108, starting from its prime factorization to extracting the square roots of perfect squares and combining the results to achieve the final simplified form.

The square root of 108 can be simplified to its most basic radical form by using the principles of prime factorization and recognizing perfect squares.

First, let's factorize 108:

• 108 is even, so divide by 2: 108 = 2 × 54
• 54 is even, so divide by 2 again: 54 = 2 × 27
• 27 is divisible by 3: 27 = 3 × 9
• 9 is a perfect square: 9 = 3 × 3

So, the prime factorization of 108 is:

108 = 2 × 2 × 3 × 3 × 3

Next, we identify the pairs of prime factors:

• Pair of 2's: 2 × 2
• Pair of 3's: 3 × 3

This can be rewritten as:

√108 = √(2 × 2 × 3 × 3 × 3) = √(2² × 3² × 3)

We can take the square roots of the pairs out of the radical:

√108 = √(2²) × √(3²) × √3 = 2 × 3 × √3

Therefore:

√108 = 6√3

So, the simplified form of the square root of 108 is 6√3.

Additionally, the decimal approximation of √108 is:

√108 ≈ 10.392

This step-by-step process helps us understand how to simplify the square root of 108 efficiently by using the method of prime factorization and recognizing perfect squares.

To find the prime factorization of 108, we start by dividing it by the smallest prime numbers:

1. 108 ÷ 2 = 54
2. 54 ÷ 2 = 27
3. 27 ÷ 3 = 9
4. 9 ÷ 3 = 3
5. 3 ÷ 3 = 1

Putting it all together, the prime factorization of 108 is:

108 = 2² × 3³

1. Start with the prime factorization of 108: \( 108 = 2^2 \times 3^3 \).
2. Identify perfect squares within the factorization: \( 36 = 2^2 \times 3^2 \).
3. Extract square roots of perfect squares: \( \sqrt{36} = 6 \) and \( \sqrt{3} \).
4. Combine the results: \( \sqrt{108} = 6\sqrt{3} \).

To identify perfect squares within the number 108:

1. Find the prime factorization of 108: \( 108 = 2^2 \times 3^3 \).
2. Look for pairs of identical prime factors that are even powers.
3. In 108, the perfect square is \( 36 = 2^2 \times 3^2 \).

To simplify the square root of 108, we begin with its prime factorization:

• First, divide 108 by 2: \( 108 \div 2 = 54 \)
• Divide 54 by 2 again: \( 54 \div 2 = 27 \)
• Now, 27 is divisible by 3: \( 27 \div 3 = 9 \)
• Finally, 9 is divisible by 3: \( 9 \div 3 = 3 \)

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

Now, rewrite the square root of 108 using its prime factors:

Apply the properties of square roots to separate perfect squares:

Calculate the square roots of the perfect squares:

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

To simplify \( \sqrt{108} \), we identify perfect squares within its prime factorization:

• The prime factorization of 108 is \( 2^2 \times 3^3 \).
• Extract the square roots of the perfect squares:

Combine the results to simplify \( \sqrt{108} \):

Now that we have identified the factors and perfect squares of 108, we can combine the results to simplify the square root of 108. Here is a step-by-step process:

1. We determined that the prime factorization of 108 is \( 2 \times 2 \times 3 \times 3 \times 3 \).
2. Next, we group the pairs of identical factors: \( (2 \times 2) \) and \( (3 \times 3) \).
3. We then apply the product rule for radicals: \( \sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} \).
4. Simplify inside the radical: \( \sqrt{(2^2) \times (3^2) \times 3} \).
5. Extract the square roots of the perfect squares: \( \sqrt{(2^2)} = 2 \) and \( \sqrt{(3^2)} = 3 \).
6. Combine these results: \( 2 \times 3 = 6 \).
7. The simplified form of \( \sqrt{108} \) is \( 6\sqrt{3} \).

Therefore, we can write:

\[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6 \times \sqrt{3} = 6\sqrt{3}
\]

This method shows how to combine the prime factors and perfect squares to achieve the simplest radical form of the square root of 108.

To find the final simplified form of the square root of 108, we will use the steps outlined in the previous sections and combine the results.

1. First, we performed the prime factorization of 108:

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

2. Next, we identified and grouped the perfect squares within the prime factors:

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

3. We then took the square roots of the perfect square groups:

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

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

4. Next, we simplified the square root expressions:

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

5. Finally, we combined the results to get the simplified form:

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

Therefore, the final simplified form of the square root of 108 is:

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

This represents the most reduced form of the square root, where 6 is the coefficient outside the radical and 3 remains inside the radical.

When simplifying the square root of 108, it's essential to avoid common errors that can lead to incorrect results. Here are some of the frequent mistakes and tips on how to avoid them:

• Incorrect Prime Factorization:

  One of the first steps in simplifying square roots is finding the prime factors of the number. For 108, the correct prime factorization is \(108 = 2^2 \times 3^3\). Ensure that you break down the number completely into its prime factors.

• Missing Perfect Squares:

  Failing to identify and extract perfect squares can result in an incorrect simplification. For example, \(108 = 36 \times 3\), and since \( \sqrt{36} = 6 \), the square root of 108 should be simplified to \(6\sqrt{3}\).

• Forgetting to Simplify Completely:

  Always simplify the square root as much as possible. If you only simplify partway, such as stopping at \( \sqrt{108} = \sqrt{4 \times 27} = 2\sqrt{27} \), you miss the further simplification to \(2 \times 3\sqrt{3} = 6\sqrt{3}\).

• Incorrect Multiplication of Square Roots:

  Remember that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Misapplying this rule can lead to errors. For instance, \( \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}\).

• Arithmetic Mistakes:

  Double-check your arithmetic when performing the factorization and multiplication steps. Small mistakes in basic calculations can throw off the entire simplification process.

By being aware of these common mistakes and following the correct steps, you can accurately simplify the square root of 108.

To solidify your understanding of simplifying the square root of 108, try solving the following practice problems. Each problem involves simplifying a square root expression, and you'll use similar steps to those we've discussed.

1. Simplify \( \sqrt{75} \)

   Steps:

   • Find the prime factorization of 75.
   • Identify the perfect square factors.
   • Rewrite the square root expression.
   • Extract the square roots of the perfect squares.
   • Combine the results to get the simplified form.

2. Simplify \( \sqrt{200} \)

   Steps:

   • Find the prime factorization of 200.
   • Identify the perfect square factors.
   • Rewrite the square root expression.
   • Extract the square roots of the perfect squares.
   • Combine the results to get the simplified form.

3. Simplify \( \sqrt{45} \)

   Steps:

   • Find the prime factorization of 45.
   • Identify the perfect square factors.
   • Rewrite the square root expression.
   • Extract the square roots of the perfect squares.
   • Combine the results to get the simplified form.

4. Simplify \( \sqrt{180} \)

   Steps:

   • Find the prime factorization of 180.
   • Identify the perfect square factors.
   • Rewrite the square root expression.
   • Extract the square roots of the perfect squares.
   • Combine the results to get the simplified form.

5. Simplify \( \sqrt{32} \)

   Steps:

   • Find the prime factorization of 32.
   • Identify the perfect square factors.
   • Rewrite the square root expression.
   • Extract the square roots of the perfect squares.
   • Combine the results to get the simplified form.

Use the steps and techniques we've covered to solve these problems. If you get stuck, refer back to the steps for simplifying square roots, and practice until you feel confident in your ability to simplify any square root expression.

Simplifying square roots, such as the square root of 108, has several practical applications in various fields. Here are some key areas where simplified square roots are commonly used:

• Mathematics:

  In algebra and geometry, simplifying square roots is essential for solving equations and understanding geometric properties. For example, simplifying √108 to 6√3 makes it easier to perform further calculations.

• Engineering:

  Engineers often simplify square roots to simplify complex calculations related to structures, materials, and electrical systems. Accurate simplifications help in precise measurements and optimizations.

• Physics:

  In physics, simplified square roots are used in formulas involving velocity, acceleration, and energy. Simplifying √108 to 6√3 can streamline computations in various physical equations.

• Computer Science:

  Computer algorithms often require simplified square roots for operations like graphics rendering, data analysis, and cryptographic functions. Simplifying square roots can enhance computational efficiency.

• Finance:

  In finance, square roots are used in risk assessments, stock price evaluations, and financial models. Simplified square roots help in making more accurate financial predictions and decisions.

Simplifying square roots is a fundamental skill that enhances problem-solving abilities and precision in various scientific, engineering, and mathematical applications. Understanding how to simplify square roots, such as converting √108 into 6√3, provides a solid foundation for tackling more complex problems in these fields.