How to Simplify the Square Root of 147: A Complete Guide

To simplify the square root of 147, we need to express it in its simplest radical form. Here is a step-by-step guide to simplify \( \sqrt{147} \) using prime factorization and the properties of square roots.

Step-by-Step Instructions

1. Factorize 147

   First, we factorize 147 into its prime factors:

   \[ 147 = 3 \times 7^2 \]

2. Rewrite the Square Root

   Next, we rewrite the square root of 147 using these factors:

   \[ \sqrt{147} = \sqrt{3 \times 7^2} \]

3. Apply the Product Rule for Radicals

   Using the product rule for radicals, which states \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we can separate the factors under the square root:

   \[ \sqrt{147} = \sqrt{3} \times \sqrt{7^2} \]

4. Simplify the Expression

   The square root of \( 7^2 \) is 7. So, we simplify the expression as follows:

   \[ \sqrt{147} = 7 \sqrt{3} \]

5. Conclusion

   Thus, the simplest radical form of \( \sqrt{147} \) is:

In decimal form, \( \sqrt{147} \approx 12.1244 \).

This process demonstrates how to break down the square root of a non-perfect square number into its simplest form using prime factorization and the properties of radicals.

Simplifying square roots is a fundamental concept in algebra that involves expressing a square root in its simplest radical form. This process helps in understanding and solving various mathematical problems more efficiently. Here, we will take a closer look at the steps involved in simplifying the square root of 147.

Let's break down the process:

1. Identify the factors of the number: List all the factors of 147, which are 1, 3, 7, 21, 49, and 147.
2. Find the perfect squares: From the list of factors, identify the perfect squares. In this case, 49 is a perfect square.
3. Divide by the perfect square: Divide 147 by the largest perfect square factor, which is 49. This gives us 147 ÷ 49 = 3.
4. Take the square root of the perfect square: Calculate the square root of 49, which is 7.
5. Combine the results: The simplified form of the square root of 147 is obtained by combining the square root of the perfect square and the remaining factor. Thus, √147 = 7√3.

By following these steps, you can simplify the square root of any number, making complex calculations more manageable and improving your problem-solving skills.

The process of simplifying the square root of 147 involves several key steps that make the calculation straightforward and clear. Let's break down the method to understand it better.

First, we recognize that the square root of 147 is expressed as:
\[ \sqrt{147} \]

To simplify this, we need to find the prime factors of 147. We start by factorizing 147:

• 147 can be divided by 3 (the smallest prime factor), giving us 49:
• 147 = 3 × 49
• Next, we factorize 49, which is 7 × 7:
• 49 = 7 × 7

Combining these, we get:
\[ 147 = 3 × 7^2 \]

Using the property of square roots, we can separate the prime factors:
\[ \sqrt{147} = \sqrt{3 × 7^2} \]

We know that:
\[ \sqrt{a × b} = \sqrt{a} × \sqrt{b} \]

Thus, we can simplify further:
\[ \sqrt{147} = \sqrt{3} × \sqrt{7^2} \]

Since the square root of \(7^2\) is 7, we have:
\[ \sqrt{147} = 7\sqrt{3} \]

Therefore, the simplified form of the square root of 147 is:
\[ 7\sqrt{3} \]

This method can be applied to any number to simplify its square root by identifying its prime factors and applying the square root properties accordingly.

In summary:

1. Factorize the number into its prime factors.
2. Group the factors into pairs of squares.
3. Pull out the square roots of the pairs.
4. Multiply the results to get the simplified form.

By following these steps, simplifying square roots becomes a systematic and easy process.

Simplifying the square root of 147 involves a few clear steps to break down the number into its simplest radical form. Here is a detailed, step-by-step guide to help you simplify the square root of 147:

1. List the Factors of 147:

   Begin by identifying all the factors of 147. The factors are:

   • 1, 3, 7, 21, 49, 147
2. Find the Largest Perfect Square:

   From the list of factors, determine the largest perfect square. In this case, it is 49.

3. Rewrite 147 as a Product:

   Express 147 as the product of the largest perfect square and another factor:

   \[ 147 = 49 \times 3 \]

4. Apply the Square Root to Both Factors:

   Using the property of square roots, rewrite the expression:

   \[ \sqrt{147} = \sqrt{49 \times 3} \]

   This can be separated into:

   \[ \sqrt{147} = \sqrt{49} \times \sqrt{3} \]

5. Calculate the Square Root of the Perfect Square:

   Find the square root of 49, which is 7:

   \[ \sqrt{49} = 7 \]

6. Combine the Results:

   Finally, multiply the results to get the simplified form:

   \[ \sqrt{147} = 7 \sqrt{3} \]

So, the simplified form of the square root of 147 is \( 7 \sqrt{3} \).

The prime factorization method is a systematic approach to simplifying the square root of a number by breaking it down into its prime factors. Here's a detailed step-by-step guide to simplify the square root of 147 using prime factorization:

1. Find Prime Factors:

   Determine the prime factors of 147. The prime factorization of 147 is:

   \[ 147 = 3 \times 7 \times 7 \] or \[ 147 = 3^1 \times 7^2 \]

2. Group the Prime Factors:

   Group the prime factors in pairs of the same number:

   \[ 147 = 3^1 \times (7^2) \]

3. Simplify Inside the Radical:

   Apply the square root to each group of prime factors:

   \[ \sqrt{147} = \sqrt{3^1 \times 7^2} = \sqrt{3} \times \sqrt{7^2} \]

4. Extract the Square Root:

   Since the square root of \(7^2\) is 7, we can simplify the expression further:

   \[ \sqrt{147} = 7 \sqrt{3} \]

Thus, the simplified form of the square root of 147 using the prime factorization method is \( 7 \sqrt{3} \).

To simplify the square root of 147, it's important to identify and use perfect squares. Perfect squares are numbers that have integer square roots. Here’s a detailed guide on how to identify perfect squares and use them in the simplification process:

1. List the factors of the number under the square root. For 147, the factors are:
   
   • 1
   • 3
   • 7
   • 21
   • 49
   • 147
2. Identify the perfect squares among these factors. For 147, the perfect squares are:
   
   • 1 (since 1² = 1)
   • 49 (since 7² = 49)
3. Choose the largest perfect square factor. In this case, it's 49.
4. Rewrite the original square root using this factor:
   \( \sqrt{147} = \sqrt{49 \times 3} \)
5. Use the property of square roots \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \):
   \( \sqrt{147} = \sqrt{49} \times \sqrt{3} \)
6. Simplify the expression by finding the square root of the perfect square:
   \( \sqrt{49} = 7 \)
7. Combine the simplified parts:
   \( \sqrt{147} = 7\sqrt{3} \)

By identifying and utilizing perfect squares, we have simplified the square root of 147 to its simplest form, \( 7\sqrt{3} \).

To simplify the square root of 147, we need to identify its prime factors and group them appropriately. This method involves breaking down the number into its prime factors and grouping them in pairs. Here’s a step-by-step guide:

1. Find the prime factors of 147.

   The prime factorization of 147 is:

   \[147 = 3 \times 7 \times 7\]

2. Group the prime factors into pairs.

   We can group the two 7s together:

   \[147 = 3 \times (7 \times 7)\]

3. Simplify the square root by taking one number from each pair.

   The square root of the paired factors is:

   \[\sqrt{147} = \sqrt{3 \times 7^2}\]

   Simplifying this, we get:

   \[\sqrt{147} = 7\sqrt{3}\]

Thus, the simplified form of the square root of 147 is:

\[7\sqrt{3}\]

To simplify the square root of 147, we follow a systematic approach to find the simplest radical form. Let's break down the process step by step:

1. Identify the prime factors of 147:

   • 147 can be expressed as 3 × 7 × 7.

2. Group the prime factors into pairs:

   • We have a pair of 7s and a single 3.

3. Extract the square root of each pair:

   • The pair of 7s can be taken out of the square root as a single 7.

4. Combine the results:

   • We are left with the square root of 3 under the radical sign.

5. Write the final simplified form:

   • Thus, the square root of 147 simplifies to 7√3.

Therefore, the simplified form of √147 is written as:

\[\sqrt{147} = 7\sqrt{3}\]

• Forgetting Perfect Squares: Always check for perfect squares within the radicand. For example, in √147, recognize that 49 (7²) is a perfect square, which can be simplified to 7√3.

• Rushing Through Simplification: Take your time to identify all prime factors correctly and simplify step by step. Missteps can lead to incorrect results.

• Ignoring Coefficients: Don’t overlook coefficients outside the square root. Ensure they are multiplied correctly after simplification.

• Mishandling Radicals: Be cautious when adding or subtracting radicals. Only like terms can be combined, e.g., 2√3 + 3√3 = 5√3, but 2√3 + 2√5 cannot be simplified further.

1. Simplify \( \sqrt{147} \).

   Solution: Identify perfect squares. \(147 = 49 \times 3\). Thus, \( \sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3} \).

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

   Solution: Factor into perfect squares. \(75 = 25 \times 3\). Therefore, \( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \).

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

   Solution: \(200 = 100 \times 2\), so \( \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \).

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

   Solution: \(50 = 25 \times 2\), hence \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \).

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

   Solution: \(98 = 49 \times 2\), resulting in \( \sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2} \).

Here are some frequently asked questions about simplifying the square root of 147:

• What is the simplest form of the square root of 147?

  The simplest form of the square root of 147 is \(7 \sqrt{3}\). This is because 147 can be factored into 49 and 3, and the square root of 49 is 7.

• How do you simplify the square root of 147 using prime factorization?

  To simplify the square root of 147 using prime factorization, follow these steps:

  1. Find the prime factors of 147: \(147 = 3 \times 7^2\).
  2. Group the factors into pairs of the same number: \(\sqrt{147} = \sqrt{3 \times 7^2}\).
  3. Take one number from each pair: \(\sqrt{7^2} = 7\).
  4. Combine the results: \(\sqrt{147} = 7 \sqrt{3}\).
• What is the decimal form of the square root of 147?

  The decimal form of the square root of 147 is approximately 12.124. This is calculated by finding the square root directly using a calculator.

• Why is the square root of 147 not a whole number?

  The square root of 147 is not a whole number because 147 is not a perfect square. A perfect square is a number that can be expressed as the square of an integer.

• What are some common mistakes to avoid when simplifying square roots?

  Common mistakes include not identifying all the factors of the number, not correctly grouping the factors, and forgetting to take the square root of the perfect squares.

Understanding how to simplify the square root of 147 provides a solid foundation for tackling similar problems involving radical expressions. By breaking down the steps and utilizing the prime factorization method, we have seen that the square root of 147 can be expressed in its simplest radical form as \( 7\sqrt{3} \). This process involves identifying perfect squares within the factors of 147 and using them to simplify the radical.

Key takeaways from simplifying the square root of 147 include:

• Identifying and using the largest perfect square factor to simplify the square root.
• Understanding the importance of prime factorization in breaking down complex radicals.
• Recognizing that the simplified form \( 7\sqrt{3} \) is more manageable and often necessary for further mathematical operations.

Through consistent practice and application of these steps, simplifying square roots becomes a straightforward and efficient process. This not only aids in mathematical problem-solving but also enhances overall numerical fluency. Keep practicing with different numbers to reinforce these concepts and improve your skills.

We hope this comprehensive guide has made the process of simplifying the square root of 147 clear and accessible. With these tools and strategies, you are well-equipped to tackle similar problems with confidence.