Simplify Square Root 147: Your Comprehensive Guide

To simplify the square root of 147, we need to find the prime factorization of 147 and then simplify the radicals.

Step-by-Step Solution

1. Prime factorize 147:

   \[
   147 \div 3 = 49 \\
   49 \div 7 = 7 \\
   7 \div 7 = 1
   \]

   So, the prime factors of 147 are \(3\) and \(7^2\).

2. Write 147 as a product of its prime factors:

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

3. Use the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\):

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

4. Simplify the square root of 147:

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

Conclusion

The simplified form of \(\sqrt{147}\) is \(7\sqrt{3}\).

Understanding how to simplify the square root of 147 is an essential mathematical skill. This guide will walk you through the process step-by-step, ensuring clarity and comprehension. By the end, you'll be able to simplify the square root of 147 with confidence, understanding the principles behind the calculations and their practical applications.

The square root of 147, represented as √147, can be simplified using the prime factorization method. This involves breaking down the number 147 into its prime factors and then applying the square root properties to simplify the expression.

Let's dive into the step-by-step process of simplifying √147:

1. Prime Factorization: Break down 147 into its prime factors. We get 147 = 3 × 7 × 7.
2. Grouping: Recognize that 7 × 7 is a perfect square, which simplifies to 7.
3. Combining: Combine the simplified square root with the remaining factor. Thus, √147 = 7√3.

The simplest radical form of the square root of 147 is 7√3. This method not only simplifies the calculation but also helps in understanding the fundamental concepts of square roots and prime factorization.

Through this guide, you will also explore common mistakes to avoid, practical applications of simplified radicals, and various practice problems to reinforce your learning.

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 multiplied by 5 equals 25.

To simplify square roots, especially for non-perfect squares like 147, we use various methods such as prime factorization. This involves breaking down the number into its prime factors to find the simplest radical form.

For 147, the prime factorization method helps us understand that it can be expressed as \( 147 = 3 \times 7 \times 7 \). By grouping the prime factors, we can simplify the square root.

Another approach involves recognizing common factors and applying the square root property to simplify the radical expression further.

The prime factorization method is an effective way to simplify the square root of a number by expressing the number as a product of its prime factors. Here’s a step-by-step guide to using this method to simplify the square root of 147:

1. Find the Prime Factors:

   Begin by breaking down 147 into its prime factors:

   • 147 ÷ 3 = 49
   • 49 ÷ 7 = 7
   • 7 ÷ 7 = 1

   Thus, the prime factorization of 147 is 3 × 7 × 7 or 3 × 7².

2. Pair the Prime Factors:

   Group the prime factors into pairs:

   • 7 × 7 forms a pair of 7s.
   • The number 3 remains unpaired.
3. Extract the Square Root:

   For each pair of identical factors, take one factor out of the square root:

   • The pair of 7s can be taken out as a single 7.
   • The unpaired 3 remains inside the square root.

   Thus, the square root of 147 can be simplified to 7√3.

In summary, the square root of 147, using the prime factorization method, simplifies to:

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

This method helps in simplifying radicals and understanding the structure of numbers.

To simplify the square root of 147, we first need to break down the number into its prime factors. Here are the steps to follow:

1. List the Factors: Start by listing all the factors of 147. The factors are:

   • 1
   • 3
   • 7
   • 21
   • 49
   • 147

2. Identify the Prime Factors: From the list of factors, we can see that 147 is divisible by 3 and 49. We then factor 49 further since it is not a prime number:

   147 = 3 × 49

   49 = 7 × 7

   So, 147 can be broken down into:

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

3. Pair the Prime Factors: For simplification, we pair the prime factors. In this case, we have one pair of 7s:

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

4. Simplify Using the Square Root Property: The square root property states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Apply this property to simplify:

   \( \sqrt{147} = \sqrt{3 \times (7 \times 7)} = \sqrt{3} \times \sqrt{7 \times 7} \)

5. Take Out the Perfect Squares: Since \(\sqrt{7 \times 7} = 7\), we can take 7 out of the radical:

   \( \sqrt{147} = 7 \times \sqrt{3} \)

6. Final Simplified Form: The square root of 147 in its simplest form is:

   \( \sqrt{147} = 7\sqrt{3} \)

The Square Root Property is a useful tool for simplifying radicals, particularly when dealing with quadratic equations. Here’s a step-by-step guide on how to use this property to simplify the square root of 147:

1. Identify the Square Root Property:

   The Square Root Property states that for any non-negative number \(k\), if \(x^2 = k\), then \(x = \pm \sqrt{k}\). This means that both the positive and negative square roots are considered.

2. Apply the Square Root Property:

   For the number 147, we start by setting up the equation:

   \[
   x^2 = 147
   \]

   Using the Square Root Property, we get:

   \[
   x = \pm \sqrt{147}
   \]

3. Simplify the Radical:

   To simplify \(\sqrt{147}\), we need to find its prime factors. From our previous section, we know:

   \[
   147 = 3 \times 49 = 3 \times 7^2
   \]

   Thus, we can rewrite the square root as:

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

   Using the property of square roots (\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)), we get:

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

   Simplifying further:

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

4. Final Expression:

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

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

By following these steps, you can simplify any similar radicals using the Square Root Property. Remember, this property is essential when dealing with quadratic equations and simplifying radicals.

Simplifying radicals involves breaking down the radicand (the number inside the square root) into its prime factors and then applying the properties of square roots to simplify the expression. Here is a step-by-step process to simplify radicals:

1. Identify the Prime Factors:

   First, find the prime factors of the radicand. For example, to simplify \( \sqrt{147} \), we break down 147 into its prime factors.

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

2. Apply the Square Root Property:

   Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to separate the factors under the square root.

   \( \sqrt{147} = \sqrt{3 \times 7^2} \)

3. Simplify the Radicals:

   Extract the square factors from under the square root. Since \( 7^2 \) is a perfect square, it can be taken out of the radical.

   \( \sqrt{147} = \sqrt{3} \times \sqrt{7^2} = \sqrt{3} \times 7 = 7\sqrt{3} \)

4. Final Result:

   Combine the simplified terms to get the final simplified radical.

   \( \sqrt{147} = 7\sqrt{3} \)

This process ensures that the radical is in its simplest form, making it easier to work with in further mathematical operations.

In this section, we will go through a detailed, step-by-step process to simplify the square root of 147. Simplifying radicals can make them easier to work with in various mathematical operations.

1. Prime Factorization:

   Start by breaking down 147 into its prime factors.

   147 can be factored as:

   \( 147 = 3 \times 49 \)

   Since 49 is a perfect square, it can be further factored:

   \( 49 = 7 \times 7 \)

   Thus, the prime factorization of 147 is:

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

2. Express the Radicand as a Product of Primes:

   Using the prime factorization, express the number under the square root as a product of primes:

   \( \sqrt{147} = \sqrt{3 \times 7^2} \)

3. Apply the Square Root Property:

   Use the property of square roots that allows us to separate the factors:

   \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)

   Apply this property to our expression:

   \( \sqrt{147} = \sqrt{3 \times 7^2} = \sqrt{3} \times \sqrt{7^2} \)

4. Simplify the Perfect Square:

   Since the square root of a perfect square (like \( 7^2 \)) is simply the base (7 in this case), we can simplify:

   \( \sqrt{7^2} = 7 \)

   So our expression becomes:

   \( \sqrt{3} \times 7 \)

   Or, more neatly written:

   \( 7 \sqrt{3} \)

5. Final Simplified Form:

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

   \( \sqrt{147} = 7\sqrt{3} \)

By following these steps, you can simplify the square root of any number by identifying and extracting perfect square factors.

When simplifying square roots, it's important to be aware of common mistakes to ensure accuracy and efficiency. Here are some pitfalls to avoid:

• Ignoring Perfect Squares:

  One common mistake is not recognizing perfect squares within the radicand. Always check if a part of the number is a perfect square, as it can simplify the process significantly. For example, in \( \sqrt{147} \), recognizing that 49 is a perfect square helps simplify it to \( 7\sqrt{3} \).

• Rushing the Process:

  Simplifying square roots requires careful attention to detail. Rushing can lead to errors. Take your time to factorize the number correctly and apply the properties of square roots methodically.

• Incorrect Prime Factorization:

  Ensure that the prime factorization of the number is accurate. Mistakes in this step can lead to incorrect simplifications. For \( \sqrt{147} \), the correct prime factors are 3 and 7, resulting in \( \sqrt{3 \times 7 \times 7} = 7\sqrt{3} \).

• Forgetting to Simplify Completely:

  After breaking down the number, ensure all possible simplifications are performed. For instance, don't stop at \( \sqrt{3 \times 49} \); simplify it further to \( 7\sqrt{3} \).

• Mixing Up Square Root and Squaring:

  Be careful not to confuse taking the square root with squaring a number. These are inverse operations, and mixing them up can lead to incorrect results.

By being mindful of these common mistakes, you can enhance your accuracy and confidence in simplifying square roots.

Simplified radicals have various practical applications in different fields. Here are some of the key areas where they are commonly used:

• Geometry:

  In geometry, simplified radicals are often used to express the lengths of sides in right triangles. For example, the Pythagorean theorem often results in square roots that need to be simplified to find the exact lengths of sides.

• Physics:

  In physics, radicals appear in formulas for calculating quantities such as wave speeds, distances in kinematic equations, and energy levels in quantum mechanics. Simplifying these radicals can make the equations more manageable and easier to interpret.

• Engineering:

  Engineering problems often involve calculations with square roots, especially in structural analysis, electrical engineering, and fluid dynamics. Simplified radicals help in presenting the results in a clear and precise manner.

• Algebra:

  In algebra, simplified radicals are crucial for solving quadratic equations and other polynomial equations. They are also used in simplifying expressions and solving radical equations.

• Trigonometry:

  Simplified radicals are frequently used in trigonometric identities and equations. For instance, the values of sine, cosine, and tangent for special angles often involve simplified square roots.

• Computer Graphics:

  In computer graphics, simplified radicals are used to calculate distances, angles, and various transformations. These calculations are essential for rendering images and animations accurately.

• Construction:

  In construction, simplified radicals can be used to calculate lengths, areas, and volumes, especially when dealing with non-standard shapes and forms. This ensures precision in measurements and material usage.

Overall, the ability to simplify radicals is a fundamental skill that aids in the clarity and accuracy of mathematical calculations across a wide range of disciplines.

To help you master the process of simplifying square roots, here are some practice problems. Follow the step-by-step simplification process outlined in the previous sections to solve these problems.

1. Simplify the square root of 75.

   Solution:

   • Step 1: Find the prime factors of 75.

     75 = 3 × 5 × 5

   • Step 2: Group the prime factors into pairs.

     (5 × 5) × 3

   • Step 3: Take the square root of each pair and move it outside the radical.

     5√3

   • Final Answer: 5√3
2. Simplify the square root of 98.

   Solution:

   • Step 1: Find the prime factors of 98.

     98 = 2 × 7 × 7

   • Step 2: Group the prime factors into pairs.

     (7 × 7) × 2

   • Step 3: Take the square root of each pair and move it outside the radical.

     7√2

   • Final Answer: 7√2
3. Simplify the square root of 200.

   Solution:

   • Step 1: Find the prime factors of 200.

     200 = 2 × 2 × 2 × 5 × 5

   • Step 2: Group the prime factors into pairs.

     (2 × 2) × (5 × 5) × 2

   • Step 3: Take the square root of each pair and move it outside the radical.

     2 × 5√2

   • Final Answer: 10√2
4. Simplify the square root of 32.

   Solution:

   • Step 1: Find the prime factors of 32.

     32 = 2 × 2 × 2 × 2 × 2

   • Step 2: Group the prime factors into pairs.

     (2 × 2) × (2 × 2) × 2

   • Step 3: Take the square root of each pair and move it outside the radical.

     2 × 2√2

   • Final Answer: 4√2
5. Simplify the square root of 245.

   Solution:

   • Step 1: Find the prime factors of 245.

     245 = 5 × 7 × 7

   • Step 2: Group the prime factors into pairs.

     (7 × 7) × 5

   • Step 3: Take the square root of each pair and move it outside the radical.

     7√5

   • Final Answer: 7√5

• What is the square root of 147?

  The square root of 147 is approximately 12.124 when expressed in decimal form. In its simplest radical form, it is \(7\sqrt{3}\).

• What is the prime factorization of 147?

  The prime factorization of 147 is \(3 \times 7^2\). This can be used to simplify the square root of 147 to \(7\sqrt{3}\).

• Is 147 a perfect square?

  No, 147 is not a perfect square. A perfect square is a number that can be expressed as the square of an integer, and there is no integer that squared gives 147.

• Is the square root of 147 a rational number?

  No, the square root of 147 is not a rational number. It is an irrational number because it cannot be expressed as a simple fraction and its decimal form is non-terminating and non-repeating.

• Can we find the square root of 147 using the repeated subtraction method?

  No, the repeated subtraction method can only be used to find the square roots of perfect squares, which 147 is not.

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

  The decimal form of the square root of 147 is approximately 12.124.

• How do you simplify the square root of 147?

  To simplify the square root of 147, you can use prime factorization. Since 147 = \(3 \times 7^2\), the square root of 147 is \(7\sqrt{3}\).

• Is -12.124 also a square root of 147?

  Yes, -12.124 is also a square root of 147 because when squared, \(-12.124 \times -12.124 = 147\).

For a deeper understanding of simplifying square roots and related concepts, consider exploring the following resources:

These resources provide comprehensive guides, practice problems, and interactive lessons to help you master the concept of simplifying square roots. Whether you are looking for detailed step-by-step instructions or additional practice problems, these websites offer valuable information and tools.

We encourage you to explore these resources to solidify your understanding and gain confidence in simplifying square roots and other mathematical concepts.