Simplify Square Root 27: A Step-by-Step Guide to Easy Simplification

To simplify the square root of 27, we need to find the prime factors of 27 and pair the factors whenever possible.

Step-by-Step Process

1. Find the prime factors of 27.
2. 27 can be written as 3 × 3 × 3.
3. Since 27 = 3³, we can rewrite it under the square root as √(3³).
4. Split the exponent inside the square root: √(3² × 3).
5. Take the square root of the perfect square: 3 × √3.

Therefore, the simplified form of √27 is:

$$\sqrt{27} = 3\sqrt{3}$$

The concept of square roots is fundamental in mathematics, providing a way to understand the inverse operation of squaring a number. 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.

Square roots are represented using the radical symbol (√). For instance, the square root of 27 is written as √27. Simplifying square roots involves expressing them in their simplest radical form. This often requires breaking down the number into its prime factors and identifying perfect squares within those factors.

Here are the basic steps to simplify a square root:

• Step 1: Find the prime factorization of the number.
• Step 2: Group the prime factors into pairs.
• Step 3: Move one factor from each pair outside the square root.
• Step 4: Multiply the factors outside the square root and simplify any remaining factors inside the radical.

Understanding these steps makes it easier to simplify any square root, including √27. In the following sections, we will explore these steps in detail and apply them to simplify √27 effectively.

Simplifying square roots is a crucial skill in mathematics, allowing us to express roots in their simplest form. Simplification involves breaking down a number into its prime factors and extracting the square roots of any perfect squares within those factors. Here's a detailed, step-by-step approach to understanding and simplifying square roots:

1. Identify the Number: Determine the number under the square root that needs simplification. For instance, we are simplifying √27.
2. Prime Factorization: Break down the number into its prime factors. For 27, the prime factorization is:
   • 27 = 3 × 3 × 3 or 3³
3. Group the Factors: Group the prime factors into pairs of two. Each pair represents a perfect square. In the case of 27, we have:
   • 3 × 3 = 9 (a perfect square)
4. Simplify the Radical: Take the square root of each perfect square and move it outside the radical. The remaining factor stays inside the radical. For √27, this looks like:
   • √(3 × 3 × 3) = √(9 × 3) = 3√3

Thus, the simplified form of √27 is 3√3. This method can be applied to any number, making it easier to work with square roots in various mathematical contexts. By mastering square root simplification, you can simplify complex expressions and solve equations more efficiently.

Simplifying the square root of 27 involves expressing the number inside the square root as a product of its prime factors and then simplifying it by extracting the perfect squares.

1. Prime Factorization:

   First, find the prime factors of 27. Break down 27 into its prime factors:

   • 27 ÷ 3 = 9
   • 9 ÷ 3 = 3
   • 3 ÷ 3 = 1

   So, the prime factorization of 27 is \( 3 \times 3 \times 3 \), or \( 3^3 \).

2. Rewrite the Square Root:

   Rewrite √27 using its prime factors:

   \( \sqrt{27} = \sqrt{3^3} \)

3. Separate the Perfect Square:

   Identify and separate the perfect square from under the square root:

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

   Since \( 3^2 \) is a perfect square, it can be simplified:

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

4. Simplify the Expression:

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

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

5. Final Simplified Form:

   Combine the simplified parts to get the final form:

   \( \sqrt{27} = 3 \sqrt{3} \)

Thus, the simplified form of \( \sqrt{27} \) is \( 3 \sqrt{3} \).

The prime factorization method is a powerful technique to simplify square roots. It involves breaking down a number into its prime factors and using these factors to simplify the square root.

1. Identify Prime Factors:

   Start by identifying the prime numbers that multiply together to give the original number. For 27, the prime factors are found as follows:

   • 27 ÷ 3 = 9
   • 9 ÷ 3 = 3
   • 3 ÷ 3 = 1

   So, the prime factors of 27 are \( 3 \times 3 \times 3 \), or \( 3^3 \).

2. Express the Number as a Product of Primes:

   Using the prime factorization, rewrite 27 as a product of its prime factors:

   \( 27 = 3 \times 3 \times 3 \) or \( 3^3 \).

3. Rewrite the Square Root:

   Rewrite the square root of 27 using its prime factorization:

   \( \sqrt{27} = \sqrt{3^3} \)

4. Simplify Using Perfect Squares:

   Identify any perfect squares in the prime factorization and simplify them:

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

   The term \( 3^2 \) is a perfect square and can be simplified:

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

5. Combine the Simplified Parts:

   Combine the simplified part with the remaining factor under the square root:

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

6. Final Simplified Form:

   Thus, the simplified form of \( \sqrt{27} \) is:

   \( \sqrt{27} = 3 \sqrt{3} \)

Using the prime factorization method, we have effectively simplified the square root of 27 to \( 3 \sqrt{3} \).

To simplify the square root of 27, we start by expressing 27 as a product of its prime factors. Prime factorization involves breaking down a number into the prime numbers that multiply together to form the original number.

1. Understanding Prime Factors:

   Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, and 7. The prime factorization process finds these prime numbers for any given number.

2. Divide by the Smallest Prime Number:

   Start dividing 27 by the smallest prime number, which is 2. Since 27 is odd, it is not divisible by 2. Move to the next smallest prime number, which is 3.

   • 27 ÷ 3 = 9

   Write down 3 as one of the prime factors.

3. Continue Dividing by Prime Numbers:

   Next, take the quotient (9) and continue the process by dividing it by the smallest prime number:

   • 9 ÷ 3 = 3

   Again, 3 is a prime factor of 27. Write down this 3.

4. Complete the Factorization:

   Finally, take the quotient (3) and divide it by the smallest prime number:

   • 3 ÷ 3 = 1

   Once we reach 1, we have completed the factorization. Write down the last 3.

5. Combine the Prime Factors:

   Collect all the prime factors written down during the division process. For 27, the prime factors are:

   \( 3 \times 3 \times 3 \) or \( 3^3 \).

Thus, the number 27 expressed as a product of its prime factors is \( 3 \times 3 \times 3 \), which is \( 3^3 \).

To simplify the square root of 27, we first need to rewrite it in a form that makes it easier to work with. The process involves breaking down the number under the square root into its prime factors.

Here's how to rewrite the square root:

1. Start with the original square root expression: \( \sqrt{27} \).
2. Perform the prime factorization of 27. The number 27 can be expressed as a product of prime factors: \( 27 = 3 \times 3 \times 3 \) or \( 27 = 3^3 \).
3. Rewrite the square root expression using the prime factorization: \( \sqrt{27} = \sqrt{3 \times 3 \times 3} \) or \( \sqrt{3^3} \).

By rewriting the square root in terms of its prime factors, we make it easier to identify and extract the perfect square, which is the next step in the simplification process.

To extract the perfect square from the square root of 27, we need to break down the number into its prime factors and identify the perfect square within it. Here’s a step-by-step process:

1. First, factorize 27 into its prime factors:

   \( 27 = 3 \times 3 \times 3 \)

2. Next, rewrite the square root of 27 using these prime factors:

   \( \sqrt{27} = \sqrt{3 \times 3 \times 3} \)

3. Group the factors into pairs to identify the perfect square. In this case, \(3 \times 3\) forms a perfect square:

   \( \sqrt{27} = \sqrt{(3 \times 3) \times 3} = \sqrt{9 \times 3} \)

4. Extract the square root of the perfect square (9) out of the radical:

   \( \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \)

5. The final simplified form of \( \sqrt{27} \) is:

   3√3

By following these steps, we have successfully extracted the perfect square and simplified the square root of 27.

To find the final simplified form of √27, follow these steps:

1. First, perform the prime factorization of 27. We get:
   • 27 = 3 × 3 × 3 = 3³
2. Next, apply the square root to the prime factorization:
   • √27 = √(3 × 3 × 3)
3. Rewrite the square root of the product:
   • √(3 × 3 × 3) = √(3² × 3)
4. Use the property of square roots that states √(a × b) = √a × √b:
   • √(3² × 3) = √(3²) × √3
5. Since the square root of 3² is 3, simplify further:
   • √(3²) × √3 = 3 × √3
6. Therefore, the simplified form of √27 is:
   • 3√3

In summary, the square root of 27 simplifies to 3√3.

Here is the final result written in MathJax:

\(\sqrt{27} = 3\sqrt{3}\)

Understanding the process of simplifying square roots can be easier with more examples. Here are a few different cases to illustrate how square roots of various numbers can be simplified:

Example 1: Simplifying √72

1. Identify the perfect square factors of 72. The largest perfect square factor is 36.
2. Express 72 as a product of 36 and 2: \(72 = 36 \times 2\).
3. Rewrite the square root: \(\sqrt{72} = \sqrt{36 \times 2}\).
4. Apply the product rule of square roots: \(\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}\).
5. Simplify the expression: \(\sqrt{36} = 6\), so \(\sqrt{72} = 6\sqrt{2}\).

Example 2: Simplifying √48

1. Identify the perfect square factors of 48. The largest perfect square factor is 16.
2. Express 48 as a product of 16 and 3: \(48 = 16 \times 3\).
3. Rewrite the square root: \(\sqrt{48} = \sqrt{16 \times 3}\).
4. Apply the product rule of square roots: \(\sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3}\).
5. Simplify the expression: \(\sqrt{16} = 4\), so \(\sqrt{48} = 4\sqrt{3}\).

Example 3: Simplifying √200

1. Identify the perfect square factors of 200. The largest perfect square factor is 100.
2. Express 200 as a product of 100 and 2: \(200 = 100 \times 2\).
3. Rewrite the square root: \(\sqrt{200} = \sqrt{100 \times 2}\).
4. Apply the product rule of square roots: \(\sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2}\).
5. Simplify the expression: \(\sqrt{100} = 10\), so \(\sqrt{200} = 10\sqrt{2}\).

Example 4: Simplifying √180

1. Identify the perfect square factors of 180. The largest perfect square factor is 36.
2. Express 180 as a product of 36 and 5: \(180 = 36 \times 5\).
3. Rewrite the square root: \(\sqrt{180} = \sqrt{36 \times 5}\).
4. Apply the product rule of square roots: \(\sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5}\).
5. Simplify the expression: \(\sqrt{36} = 6\), so \(\sqrt{180} = 6\sqrt{5}\).

Example 5: Simplifying √54a¹⁰b¹⁶c⁷

1. Break down each variable and number into their prime factors.
2. Express the number and variables inside the square root: \(\sqrt{54a^{10}b^{16}c^{7}} = \sqrt{54} \times \sqrt{a^{10}} \times \sqrt{b^{16}} \times \sqrt{c^{7}}\).
3. Simplify each part:
   • \(\sqrt{54} = 3\sqrt{6}\) (since \(54 = 9 \times 6\) and \(\sqrt{9} = 3\))
   • \(\sqrt{a^{10}} = a^{5}\)
   • \(\sqrt{b^{16}} = b^{8}\)
   • \(\sqrt{c^{7}} = c^{3}\sqrt{c}\) (since \(c^{7} = c^{6} \times c\) and \(\sqrt{c^{6}} = c^{3}\))
4. Combine the simplified parts: \(\sqrt{54a^{10}b^{16}c^{7}} = 3a^{5}b^{8}c^{3}\sqrt{6c}\).

When simplifying square roots, it's easy to make mistakes that can lead to incorrect answers. Here are some common pitfalls and tips to avoid them:

• Forgetting to Check for Perfect Squares:

  Always check if the number under the square root is a perfect square. For example, if you have √36, recognize that 36 is a perfect square (6²), so √36 = 6.

• Incorrect Factorization:

  Ensure that you factor the number correctly into its prime factors. For example, for √72, the correct factorization is 72 = 2³ × 3², leading to √72 = √(2³ × 3²).

• Ignoring Pairs of Factors:

  When simplifying, look for pairs of factors inside the square root. For example, in √72 = √(2³ × 3²), you can pair the factors 3² and take them outside the root as a single 3, resulting in 3√8.

• Rushing Through the Process:

  Simplifying square roots requires careful steps. Rushing can lead to errors. Take your time to factorize and simplify step by step.

• Incorrect Handling of Variables:

  When dealing with variables under the square root, remember the property √(x²) = |x|, especially if x can be negative. This ensures your answer is always positive.

By keeping these common mistakes in mind and practicing regularly, you can master the art of simplifying square roots with confidence!

Practicing square root simplifications can help solidify your understanding and ensure you avoid common mistakes. Here are some practice problems to try:

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

   1. Factor 45 into prime factors: \( 45 = 3^2 \times 5 \)
   2. Rewrite using the square root property: \( \sqrt{45} = \sqrt{3^2 \times 5} \)
   3. Extract the perfect square: \( 3\sqrt{5} \)

   Answer: \( 3\sqrt{5} \)

2. Simplify \( \sqrt{50} \)

   1. Factor 50 into prime factors: \( 50 = 2 \times 5^2 \)
   2. Rewrite using the square root property: \( \sqrt{50} = \sqrt{2 \times 5^2} \)
   3. Extract the perfect square: \( 5\sqrt{2} \)

   Answer: \( 5\sqrt{2} \)

3. Simplify \( \sqrt{72} \)

   1. Factor 72 into prime factors: \( 72 = 2^3 \times 3^2 \)
   2. Rewrite using the square root property: \( \sqrt{72} = \sqrt{2^3 \times 3^2} \)
   3. Extract the perfect square: \( 3\sqrt{8} \)
   4. Further simplify \( \sqrt{8} \): \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \)
   5. Combine: \( 3 \times 2\sqrt{2} = 6\sqrt{2} \)

   Answer: \( 6\sqrt{2} \)

4. Simplify \( \sqrt{98} \)

   1. Factor 98 into prime factors: \( 98 = 2 \times 7^2 \)
   2. Rewrite using the square root property: \( \sqrt{98} = \sqrt{2 \times 7^2} \)
   3. Extract the perfect square: \( 7\sqrt{2} \)

   Answer: \( 7\sqrt{2} \)

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

   1. Factor 200 into prime factors: \( 200 = 2^3 \times 5^2 \)
   2. Rewrite using the square root property: \( \sqrt{200} = \sqrt{2^3 \times 5^2} \)
   3. Extract the perfect square: \( 5\sqrt{8} \)
   4. Further simplify \( \sqrt{8} \): \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \)
   5. Combine: \( 5 \times 2\sqrt{2} = 10\sqrt{2} \)

   Answer: \( 10\sqrt{2} \)

These practice problems will help you become more confident in simplifying square roots. Remember to always factorize completely and look for perfect squares to simplify your work.

In this comprehensive guide, we have explored the process of simplifying the square root of 27 in a detailed and systematic manner. Let's recap the key points:

• We started by understanding the basics of square roots and their significance in mathematics.
• The process of simplification was introduced, emphasizing the need to break down the number under the radical into its prime factors.
• We illustrated the step-by-step simplification of √27 using the prime factorization method:
• Expressing 27 as a product of its prime factors: \( 27 = 3^3 \)
• Rewriting the square root: \( \sqrt{27} = \sqrt{3^3} \)
• Extracting the perfect square: \( \sqrt{3^3} = 3\sqrt{3} \)
• The final simplified form of √27 is \( 3\sqrt{3} \).
• We also explored other examples of square root simplifications to reinforce the concept.
• Common mistakes to avoid were highlighted to ensure accuracy in simplification.
• Practice problems were provided to help you apply what you have learned.

Understanding the simplification of square roots, like √27, not only enhances your mathematical skills but also builds a solid foundation for tackling more complex algebraic expressions. Practice regularly and avoid common errors to master this fundamental concept.

With these steps and insights, you can confidently simplify square roots and apply these techniques to a variety of mathematical problems. Keep practicing and exploring the fascinating world of mathematics!