Simplify Square Root 216: Your Comprehensive Guide to Simplifying √216

Simplifying square roots involves finding the prime factorization of the number under the radical and then simplifying the radical by grouping pairs of the same factors.

Steps to Simplify √216

1. Find the prime factorization of 216.
2. Identify pairs of prime factors.
3. Simplify the square root by taking one number from each pair outside the radical.

Prime Factorization of 216

First, we find the prime factors of 216:

• 216 is divisible by 2: 216 ÷ 2 = 108
• 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 a prime number

Thus, the prime factorization of 216 is: \(216 = 2^3 \times 3^3\)

Simplifying the Square Root

Next, we simplify the square root by grouping the prime factors:

\[\sqrt{216} = \sqrt{2^3 \times 3^3}\]

Group the factors into pairs:

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

Combine the simplified factors:

\[\sqrt{216} = 2 \times 3 \times \sqrt{6} = 6\sqrt{6}\]

Final Result

Thus, the simplified form of \(\sqrt{216}\) is:

\[\sqrt{216} = 6\sqrt{6}\]

1. Introduction to Simplifying Square Root 216
2. Understanding the Factors of 216
3. Prime Factorization of 216
4. Methods to Simplify Square Root 216
5. Step-by-Step Guide to Simplifying √216
6. Common Techniques for Simplifying Square Roots
7. Using Factor Trees for Square Root Simplification
8. Examples of Simplifying √216
9. Practice Problems and Solutions
10. Conclusion: Mastering the Simplification of √216

Simplifying the square root of 216 involves expressing it in its simplest radical form. The process is straightforward and involves breaking down the number into its prime factors, identifying the perfect squares, and simplifying accordingly.

Here is a step-by-step approach:

1. List the factors of 216:
   
   • 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
2. Identify the perfect squares from the list of factors:
   
   • 1, 4, 9, 36
3. Divide 216 by the largest perfect square identified:
   
   • 216 / 36 = 6
4. Calculate the square root of the largest perfect square:
   
   • √36 = 6
5. Combine these results to express √216 in its simplest form:
   
   • √216 = 6√6

This method of simplification helps in various mathematical computations and is a fundamental skill in algebra. For example, knowing that √216 can be simplified to 6√6 allows for easier calculations in more complex equations.

By following these steps, you can simplify the square root of 216 efficiently:

1. Prime Factorization: Break down 216 into its prime factors: 2³ × 3³
2. Grouping Factors: Group the factors into pairs of squares: (2²) × (3²) × 6
3. Applying the Square Root: Simplify the pairs: √216 = √[(2²) × (3²) × 6] = 2 × 3 × √6
4. Final Simplification: Combine the results: 2 × 3 × √6 = 6√6

Therefore, the simplified form of the square root of 216 is 6√6, which can also be approximated as 14.697 in decimal form.

Simplifying the square root of 216 involves breaking it down into its prime factors and using properties of square roots to find the simplest radical form. Here are the methods step-by-step:

Method 1: Prime Factorization

1. Prime Factorize 216:

   216 can be factored into its prime factors: 216 = 2 × 2 × 2 × 3 × 3 × 3, or written as 2³ × 3³.

2. Group the Prime Factors:

   Group the factors into pairs of identical factors: (2 × 2) × (3 × 3) × 2 × 3.

3. Simplify the Square Root:

   Take the square root of each pair:
   

   
   
   • √(2 × 2) = 2
   
   
   • √(3 × 3) = 3
   
   
   
   Multiply the results and leave the remaining factors under the square root:

   2 × 3 × √(2 × 3) = 6√6

Method 2: Using Perfect Squares

1. Identify Perfect Squares:

   List the factors of 216 and identify the perfect squares: 1, 4, 9, 36.

2. Divide by the Largest Perfect Square:

   Divide 216 by the largest perfect square (36):

   216 ÷ 36 = 6

3. Combine the Results:

   Take the square root of the perfect square and multiply it by the square root of the remaining factor:

   √36 = 6, so √216 = 6√6

Method 3: Factor Tree

1. Create a Factor Tree:

   Break down 216 into a factor tree until all branches end in prime numbers:

   
   

   
   
   • 216
   
   
   • / \
     
     
     
     • 2
     
     
     • 108
     
     
     
     
   
   
   • / \
     
     
     
     • 2
     
     
     • 54
     
     
     
     
   
   
   • / \
     
     
     
     • 2
     
     
     • 27
     
     
     
     
   
   
   • / \
     
     
     
     • 3
     
     
     • 9
     
     
     
     
   
   
   • / \
     
     
     
     • 3
     
     
     • 3
     
     
     
     
   
   
   
   


2. Pair the Primes:

   Pair the primes and move them out of the radical:

   (2 × 2) × (3 × 3) × 2 × 3 = 6√6

Conclusion

By using prime factorization, identifying perfect squares, or employing a factor tree, the square root of 216 can be simplified to 6√6. Each method demonstrates the underlying principles of breaking down a number into manageable components and simplifying the expression systematically.

To simplify the square root of 216, we follow a series of systematic steps:

1. Factorize 216

   Begin by breaking down 216 into its prime factors:

   
   \[ 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \]

2. Group the Prime Factors

   Group the prime factors into pairs of identical factors:

   
   \[ 216 = (2 \times 2) \times (3 \times 3) \times 2 \times 3 \]

3. Rewrite Using Radical Notation

   Rewrite the expression under the square root sign using the pairs of factors:

   
   \[ \sqrt{216} = \sqrt{(2^2) \times (3^2) \times 6} \]

4. Simplify the Square Roots

   Extract the pairs of factors from under the radical sign:

   
   \[ \sqrt{216} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{6} \]

   This simplifies to:

   
   \[ \sqrt{216} = 2 \times 3 \times \sqrt{6} \]

5. Combine the Results

   Finally, multiply the numbers outside the square root:

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

Thus, the simplified form of the square root of 216 is:

\[ \sqrt{216} = 6 \sqrt{6} \]

The number 216 is a composite number, which means it has more than two factors. To understand the factors of 216, we need to look at both its prime factorization and the pairs of numbers that multiply to give 216.

Prime Factorization of 216

Prime factorization involves breaking down a number into its smallest prime factors. For 216, the prime factorization is:

\[ 216 = 2^3 \times 3^3 \]

This means that 216 can be expressed as the product of the primes 2 and 3, each raised to the power of 3.

Factor Tree of 216

A factor tree helps visualize the prime factorization of 216:

Factors of 216

The factors of 216 are the numbers that divide 216 exactly without leaving a remainder. These factors can be paired such that the product of each pair is 216. Here are the factor pairs of 216:

• 1 and 216
• 2 and 108
• 3 and 72
• 4 and 54
• 6 and 36
• 8 and 27
• 9 and 24
• 12 and 18

Each pair multiplies to give 216:

\[ 1 \times 216 = 216 \]

\[ 2 \times 108 = 216 \]

\[ 3 \times 72 = 216 \]

\[ 4 \times 54 = 216 \]

\[ 6 \times 36 = 216 \]

\[ 8 \times 27 = 216 \]

\[ 9 \times 24 = 216 \]

\[ 12 \times 18 = 216 \]

By understanding these factors and their pairs, we can better comprehend the structure of the number 216 and how it can be simplified or broken down for various mathematical purposes.

Simplifying square roots involves several techniques that make it easier to work with these expressions. Here are some common methods:

1. Prime Factorization

Prime factorization involves breaking down the number under the square root into its prime factors.

• Find the prime factors of the number.
• Group the prime factors into pairs.
• Take one factor from each pair outside the square root.

Example:

\(\sqrt{72}\)

1. Prime factors of 72: \(72 = 2 \times 2 \times 2 \times 3 \times 3\).
2. Group into pairs: \((2 \times 2) \times (3 \times 3) \times 2\).
3. Simplify: \(2 \times 3 \times \sqrt{2} = 6\sqrt{2}\).

2. Using Perfect Squares

Identify perfect squares that are factors of the number under the square root.

• Find the largest perfect square factor.
• Rewrite the square root as a product of the square root of the perfect square and the remaining factor.

Example:

\(\sqrt{50}\)

1. Largest perfect square factor: \(25\).
2. Rewrite: \(\sqrt{50} = \sqrt{25 \times 2}\).
3. Simplify: \(\sqrt{25} \times \sqrt{2} = 5\sqrt{2}\).

3. Simplifying Fractions

When dealing with fractions under a square root, simplify each part separately.

• Apply the rule: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).
• Simplify both the numerator and the denominator.

Example:

\(\sqrt{\frac{16}{9}}\)

1. Rewrite: \(\frac{\sqrt{16}}{\sqrt{9}}\).
2. Simplify: \(\frac{4}{3}\).

4. Combining Like Terms

If there are multiple square roots in an expression, combine like terms when possible.

• Simplify each square root separately.
• Combine the simplified forms if they are like terms.

Example:

Simplify: \(2\sqrt{3} + 4\sqrt{3}\)

1. Both terms have \(\sqrt{3}\) as a common factor.
2. Combine: \(2\sqrt{3} + 4\sqrt{3} = 6\sqrt{3}\).

5. Rationalizing the Denominator

Rationalizing involves eliminating the square root from the denominator of a fraction.

• Multiply both the numerator and the denominator by the conjugate if necessary.
• Simplify the resulting expression.

Example:

Simplify: \(\frac{1}{\sqrt{2}}\)

1. Multiply by the conjugate: \(\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\).

Factor trees are a helpful tool for simplifying square roots. They allow us to break down a number into its prime factors, making it easier to identify pairs of factors that can be simplified. Here's a step-by-step guide to using factor trees for square root simplification:

1. Identify the number to be simplified:

   Let's simplify \( \sqrt{216} \).

2. Create a factor tree:

   Start by dividing 216 by its smallest prime factor, which is 2. Continue factoring until all factors are prime numbers.

   • 216 ÷ 2 = 108
   • 108 ÷ 2 = 54
   • 54 ÷ 2 = 27
   • 27 ÷ 3 = 9
   • 9 ÷ 3 = 3
   • 3 ÷ 3 = 1

   The prime factors of 216 are: \(2 \times 2 \times 2 \times 3 \times 3 \times 3\).

3. Pair the prime factors:

   Group the prime factors into pairs. Each pair of the same number can be taken out of the square root as a single number.

   • Pairs: \( (2 \times 2) \), \( (3 \times 3) \)
   • Remaining factor: \(3\)
4. Simplify the expression:

   Each pair is simplified as follows: \( \sqrt{2 \times 2} = 2 \) and \( \sqrt{3 \times 3} = 3 \).

   The simplified expression is: \(2 \times 3 \times \sqrt{3} = 6\sqrt{3}\).

Thus, the simplified form of \( \sqrt{216} \) is \( 6\sqrt{3} \).

Using factor trees helps visually break down the number and easily identify pairs for simplification, making the process straightforward and manageable.

Let's explore some examples to understand how to simplify √216 using different methods. Here, we will use prime factorization and the product rule.

Example 1: Using Prime Factorization

1. Find the prime factors of 216:

   \(216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3\)

2. Group the factors into pairs:

   \(216 = (2 \times 2) \times (3 \times 3) \times 6\)

3. Take one factor from each pair out of the square root:

   \(\sqrt{216} = \sqrt{(2^2 \times 3^2) \times 6} = 2 \times 3 \times \sqrt{6} = 6\sqrt{6}\)

Example 2: Using the Product Rule

1. Rewrite the radicand as a product of perfect squares:

   \(\sqrt{216} = \sqrt{36 \times 6}\)

2. Separate the square root of the product into the product of square roots:

   \(\sqrt{36 \times 6} = \sqrt{36} \times \sqrt{6}\)

3. Simplify the perfect square:

   \(\sqrt{36} \times \sqrt{6} = 6 \times \sqrt{6}\)

Example 3: Using Factor Trees

1. Create a factor tree for 216:

   
   

   
   
   • 216
   
   
   
   
   • 2
   
   
   
   
   • 108
   
   
   
   
   • 2
   
   
   
   
   • 54
   
   
   
   
   • 2
   
   
   
   
   • 27
   
   
   
   
   • 3
   
   
   
   
   • 9
   
   
   
   
   • 3
   
   
   
   
   • 3
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   


2. Write down the prime factorization:

   \(216 = 2^3 \times 3^3\)

3. Pair the factors and simplify:

   \(\sqrt{216} = \sqrt{(2^2 \times 3^2) \times (2 \times 3)} = 6\sqrt{6}\)

By following these examples, you can simplify \(\sqrt{216}\) to \(6\sqrt{6}\).

Here are some practice problems to help you master the process of simplifying square roots, including the square root of 216. Each problem includes a step-by-step solution.

1. Problem 1: Simplify \( \sqrt{72} \)

   1. Prime factorize 72: \( 72 = 2^3 \times 3^2 \)
   2. Group the factors into pairs: \( \sqrt{2^3 \times 3^2} = \sqrt{(2^2 \times 3^2) \times 2} \)
   3. Simplify the pairs: \( \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2} = 2 \times 3 \times \sqrt{2} \)
   4. Combine the results: \( 6\sqrt{2} \)

2. Problem 2: Simplify \( \sqrt{98} \)

   1. Prime factorize 98: \( 98 = 2 \times 7^2 \)
   2. Group the factors into pairs: \( \sqrt{2 \times 7^2} = \sqrt{7^2} \times \sqrt{2} \)
   3. Simplify the pairs: \( 7\sqrt{2} \)

3. Problem 3: Simplify \( \sqrt{128} \)

   1. Prime factorize 128: \( 128 = 2^7 \)
   2. Group the factors into pairs: \( \sqrt{2^7} = \sqrt{(2^6) \times 2} = \sqrt{(2^3)^2 \times 2} \)
   3. Simplify the pairs: \( 2^3 \sqrt{2} = 8\sqrt{2} \)

4. Problem 4: Simplify \( \sqrt{200} \)

   1. Prime factorize 200: \( 200 = 2^3 \times 5^2 \)
   2. Group the factors into pairs: \( \sqrt{2^3 \times 5^2} = \sqrt{2^2 \times 5^2 \times 2} \)
   3. Simplify the pairs: \( \sqrt{2^2} \times \sqrt{5^2} \times \sqrt{2} = 2 \times 5 \times \sqrt{2} \)
   4. Combine the results: \( 10\sqrt{2} \)

5. Problem 5: Simplify \( \sqrt{216} \)

   1. Prime factorize 216: \( 216 = 2^3 \times 3^3 \)
   2. Group the factors into pairs: \( \sqrt{2^3 \times 3^3} = \sqrt{(2^2 \times 3^2) \times 6} \)
   3. Simplify the pairs: \( \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{6} = 2 \times 3 \times \sqrt{6} \)
   4. Combine the results: \( 6\sqrt{6} \)

Practice these problems to become more comfortable with simplifying square roots. Make sure to prime factorize the number under the square root, group the factors into pairs, simplify the pairs, and then combine the results.

Simplifying the square root of 216 involves understanding and applying a few fundamental techniques. By mastering these methods, you can confidently simplify not only √216 but also other square roots.

First, recognizing the importance of prime factorization allows you to break down any number into its prime components. For 216, this means:

1. Finding the prime factors: \(2^3 \times 3^3\).
2. Using these factors to simplify the square root: \(\sqrt{216} = \sqrt{2^3 \times 3^3} = 6\sqrt{6}\).

Second, applying the technique of using the greatest perfect square factor helps in simplification. The largest perfect square factor of 216 is 36, leading to:

1. Breaking down 216 into \(36 \times 6\).
2. Simplifying the square root: \(\sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6}\).

Additionally, visualizing the factor tree method and understanding the product rule for radicals further solidifies your ability to simplify square roots. Using these techniques, you can approach square root problems methodically and efficiently.

Ultimately, mastering these methods provides a solid foundation for tackling more complex mathematical problems, enhancing your overall math proficiency. With practice, simplifying square roots becomes a straightforward and intuitive process.

By following these steps and continuing to practice, you'll find that simplifying square roots, including \(\sqrt{216}\), becomes second nature.