Simplify Square Root of 175: A Comprehensive Step-by-Step Guide

In this section, we will simplify the square root of 175 using the prime factorization method.

Understanding the Process

To simplify a square root, we need to find the prime factors of the number and then pair them to move outside the radical. Here’s the step-by-step process:

1. Find the prime factorization of 175.
2. Group the prime factors into pairs.
3. Move each pair of prime factors outside the square root.
4. Multiply the factors outside the radical.
5. Simplify the expression inside the square root if possible.

Step-by-Step Solution

• Find the prime factorization of 175:

  \( 175 = 5 \times 5 \times 7 \)

• Group the prime factors:

  \( 175 = 5^2 \times 7 \)

• Move each pair outside the square root:

  \( \sqrt{175} = \sqrt{5^2 \times 7} \)

• Simplify the expression:

  \( \sqrt{175} = 5\sqrt{7} \)

Final Result

The simplified form of the square root of 175 is:

Conclusion

By following these steps, we can simplify any square root expression using prime factorization. Practice with different numbers to become more comfortable with this process.

The concept of square roots is fundamental in mathematics, often appearing in various areas such as algebra, geometry, and calculus. Understanding square roots is crucial for solving quadratic equations, working with exponential functions, and simplifying radical expressions.

Square roots are represented by the radical symbol (√) and indicate a value that, when multiplied by itself, yields the original number. For instance, the square root of 25 is 5 because 5 × 5 = 25. Similarly, the square root of 36 is 6 because 6 × 6 = 36.

When dealing with non-perfect squares, the square root can often be simplified to its simplest radical form. For example, the square root of 175 can be simplified as follows:

1. List the factors of 175: 1, 5, 7, 25, 35, 175.
2. Identify the perfect squares: 1, 25.
3. Divide 175 by the largest perfect square: 175 ÷ 25 = 7.
4. Calculate the square root of the largest perfect square: √25 = 5.
5. Combine the results: √175 = 5√7.

Thus, the simplified form of the square root of 175 is 5√7. This method of simplification helps in better understanding and solving mathematical problems efficiently.

Square roots can also be approximated using the long division method or by using calculators for precise decimal values. For instance, the square root of 175 is approximately 13.2287.

Prime factorization is the process of expressing a number as the product of its prime factors. This method is particularly useful in simplifying square roots, as it helps identify the perfect squares within a number. Let's explore the prime factorization of 175 step by step:

1. List the Factors:

   Identify all factors of 175. The factors are: 1, 5, 7, 25, 35, and 175.

2. Identify Prime Factors:

   From the list, determine the prime factors. The prime factors of 175 are 5 and 7.

3. Group Prime Factors:

   Express 175 as a product of its prime factors:
   
   
   \( 175 = 5 \times 5 \times 7 \)

4. Write in Exponential Form:

   Convert the product into exponential form:
   
   
   \( 175 = 5^2 \times 7^1 \)

5. Simplify the Square Root:

   Apply the square root to each group:
   
   
   \( \sqrt{175} = \sqrt{5^2 \times 7} \)
   
   
   \( \sqrt{175} = \sqrt{5^2} \times \sqrt{7} \)
   
   
   \( \sqrt{175} = 5 \sqrt{7} \)

Therefore, the simplified form of \( \sqrt{175} \) is \( 5 \sqrt{7} \).

The square root of 175 is a value that, when multiplied by itself, gives the number 175. To find the square root of 175, we use the prime factorization method. This method helps us express the square root in its simplest radical form.

Here is the step-by-step process:

1. First, find the prime factors of 175. The prime factors of 175 are 5 and 7, so we can write 175 as \(5^2 \times 7\).
2. Next, use these factors to simplify the square root. We write the square root of 175 as \(\sqrt{175} = \sqrt{5^2 \times 7}\).
3. Separate the square root into the product of square roots: \(\sqrt{5^2} \times \sqrt{7}\).
4. Since \(\sqrt{5^2}\) is 5, we simplify this to \(5\sqrt{7}\).

Thus, the square root of 175 simplified is \(5\sqrt{7}\). In decimal form, the square root of 175 is approximately 13.2287.

This simplified form is particularly useful in various mathematical contexts, such as solving equations and simplifying expressions involving square roots.

When simplifying square roots, the goal is to express the root in its simplest radical form. This involves breaking down the number into its prime factors and identifying perfect squares. Here, we will detail the methods used to simplify the square root of 175.

1. Prime Factorization Method:
   • Step 1: Find the prime factors of 175. The prime factorization of 175 is \( 5^2 \times 7 \).
   • Step 2: Group the factors into pairs of equal factors. Here, \( 5^2 \) is a pair.
   • Step 3: Take one factor from each pair outside the radical. For \( 5^2 \), take 5 outside.
   • Step 4: Multiply the factors outside the radical by the remaining factors inside the radical. Therefore, \( \sqrt{175} = 5\sqrt{7} \).
2. Using the Long Division Method:
   • Step 1: Pair the digits of the number starting from the right. For 175, we pair it as 1|75.
   • Step 2: Find a number whose square is less than or equal to the first pair. Here, 1 fits, so we subtract 1 from 1, leaving 0.
   • Step 3: Bring down the next pair (75), giving us 075. Double the divisor (which is 1), giving us 2, and find a number to place next to 2 to multiply by itself to get less than or equal to 75. Here, 3 works (since 23*3 = 69).
   • Step 4: Subtract and bring down pairs until you reach the desired decimal places. For √175, continuing this process yields approximately 13.228.

By using these methods, you can simplify the square root of 175 to its exact form, \( 5\sqrt{7} \), or approximate it to a decimal, 13.228.

Simplifying the square root of 175 involves breaking it down into its prime factors and simplifying the expression under the radical sign. Follow these detailed steps to achieve the simplest form of the square root of 175:

1. List Factors: First, list all the factors of 175. They are: 1, 5, 7, 25, 35, 175.

2. Identify Perfect Squares: From the list of factors, identify the perfect squares. Here, the perfect squares are 1 and 25.

3. Divide by the Largest Perfect Square: Divide 175 by the largest perfect square identified, which is 25:

   \[ \frac{175}{25} = 7 \]

4. Square Root of the Perfect Square: Calculate the square root of the largest perfect square:

   \[ \sqrt{25} = 5 \]

5. Combine Results: Combine the results of steps 3 and 4 to get the simplified form of the square root of 175:

   \[ \sqrt{175} = 5 \sqrt{7} \]

The square root of 175 in its simplest form is \(5 \sqrt{7}\). This method ensures that the expression under the radical sign is as simplified as possible, making calculations easier and more manageable.

The prime factorization of a number involves expressing the number as a product of its prime factors. To simplify the square root of 175, we need to find its prime factors.

1. First, identify the smallest prime number that divides 175. In this case, 175 is divisible by 5.
2. Divide 175 by 5: \(175 \div 5 = 35\).
3. Next, factorize 35. It is also divisible by 5: \(35 \div 5 = 7\).
4. Finally, 7 is a prime number.

Therefore, the prime factorization of 175 is:

\[ 175 = 5^2 \times 7 \]

Using this factorization, we can simplify the square root of 175:

\[ \sqrt{175} = \sqrt{5^2 \times 7} \]

Since the square root of a product is the product of the square roots, we have:

\[ \sqrt{5^2 \times 7} = \sqrt{5^2} \times \sqrt{7} \]

The square root of \(5^2\) is 5:

\[ \sqrt{5^2} = 5 \]

Thus, the simplified form of the square root of 175 is:

\[ \sqrt{175} = 5\sqrt{7} \]

This is the simplest radical form of the square root of 175.

To simplify the square root of 175, we need to break down 175 into its prime factors and then group these factors appropriately. Here is a step-by-step guide:

1. Prime Factorization: First, we need to find the prime factors of 175. The prime factorization of 175 is:

   
   \[
   175 = 5^2 \times 7
   \]

2. Grouping the Factors: Group the prime factors in pairs. In our case, we have:

   • \(5^2\): This is a pair of 5s.
   • \(7\): This remains as a single 7.

3. Moving the Pairs Outside the Radical: For each pair of the same factor, we take one factor out of the square root. Since we have one pair of 5s, we take a single 5 out of the radical:

   
   \[
   \sqrt{175} = \sqrt{5^2 \times 7} = 5\sqrt{7}
   \]

By following these steps, we have successfully simplified the square root of 175 to its simplest form:

\[
\sqrt{175} = 5\sqrt{7}
\]

To simplify the square root of 175, we need to move pairs of prime factors outside the radical. Here's a detailed step-by-step process:

1. First, we factorize 175 into its prime factors:

   
   \( 175 = 5 \times 5 \times 7 \)
   
   or
   
   \( 175 = 5^2 \times 7 \)

2. Next, we look for pairs of prime factors. In this case, we have a pair of 5s:

   
   \( 175 = (5 \times 5) \times 7 \)

3. We then take one 5 out of the radical, since \(\sqrt{5^2} = 5\):

   
   \( \sqrt{175} = \sqrt{5^2 \times 7} = 5\sqrt{7} \)

By following these steps, we have successfully simplified the square root of 175. The final simplified form is:

\( \sqrt{175} = 5\sqrt{7} \)

After following the steps to factor and simplify the square root of 175, we arrive at its final simplified form. Here’s a detailed breakdown of the process:

1. Prime Factorization: First, we perform the prime factorization of 175:

   \( 175 = 5^2 \times 7 \)

2. Grouping the Factors: We group the factors inside the radical, focusing on pairs of the same number:

   \( \sqrt{175} = \sqrt{5^2 \times 7} \)

3. Moving Pairs Outside the Radical: We move pairs of factors outside the radical. Each pair of the same number inside the radical becomes a single number outside:

   \( \sqrt{5^2 \times 7} = 5\sqrt{7} \)

Thus, the final simplified form of the square root of 175 is:

\( \sqrt{175} = 5\sqrt{7} \)

This means that \( \sqrt{175} \) can be expressed as \( 5\sqrt{7} \) in its simplest radical form. For decimal representation, \( \sqrt{175} \) is approximately \( 13.2287 \).

When simplifying the square root of 175, there are several common mistakes that can occur. By being aware of these errors, you can ensure a more accurate simplification process. Here are some of the most frequent mistakes to avoid:

• Not Identifying Prime Factors Correctly:

  Always ensure you correctly identify the prime factors of 175. The correct prime factorization of 175 is \(5 \times 5 \times 7\). Incorrect factorization can lead to errors in the simplification process.

• Ignoring Pairs of Prime Factors:

  One of the key steps in simplifying square roots is recognizing pairs of prime factors. In the case of 175, the pair of 5s should be moved outside the radical. Missing this step will result in an incorrect simplified form.

• Forgetting to Simplify Completely:

  After identifying and moving pairs outside the radical, ensure the expression is in its simplest form. For example, \(\sqrt{175}\) simplifies to \(5\sqrt{7}\). Leaving it as \(\sqrt{25 \times 7}\) is not fully simplified.

• Not Checking for Perfect Squares:

  Before starting the simplification, always check if the number itself or any factors are perfect squares. This can save time and simplify the process. For 175, recognizing 25 as a perfect square helps quickly simplify the expression.

• Misplacing Decimal Points:

  When converting to decimal form for approximation, ensure accuracy in your calculations. For instance, the square root of 175 is approximately 13.22876. Incorrect decimal placement can lead to significant errors.

By keeping these common mistakes in mind, you can avoid errors and achieve accurate simplification of the square root of 175.

Below are several practice problems to help you master the simplification of square roots. Follow the steps to simplify each radical expression.

1. Simplify \( \sqrt{175} \):
   • Find the prime factors of 175: \( 175 = 5 \times 5 \times 7 \)
   • Group the prime factors: \( 5 \times 5 = 25 \)
   • Move the perfect square out of the radical: \( \sqrt{175} = 5\sqrt{7} \)
2. Simplify \( \sqrt{208} \):
   • Find the prime factors of 208: \( 208 = 2 \times 2 \times 2 \times 2 \times 13 \)
   • Group the prime factors: \( 2 \times 2 = 4 \)
   • Move the perfect square out of the radical: \( \sqrt{208} = 4\sqrt{13} \)
3. Simplify \( \sqrt{294} \):
   • Find the prime factors of 294: \( 294 = 2 \times 3 \times 7 \times 7 \)
   • Group the prime factors: \( 7 \times 7 = 49 \)
   • Move the perfect square out of the radical: \( \sqrt{294} = 7\sqrt{6} \)
4. Simplify \( \sqrt{648} \):
   • Find the prime factors of 648: \( 648 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \)
   • Group the prime factors: \( 2 \times 2 = 4 \) and \( 3 \times 3 \times 3 = 27 \)
   • Move the perfect square out of the radical: \( \sqrt{648} = 18\sqrt{2} \)
5. Simplify \( \sqrt{2{,}448} \):
   • Find the prime factors of 2448: \( 2{,}448 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 17 \)
   • Group the prime factors: \( 2 \times 2 = 4 \), \( 3 \times 3 = 9 \)
   • Move the perfect square out of the radical: \( \sqrt{2{,}448} = 12\sqrt{17} \)

Practice these problems and check your answers to ensure you understand the process of simplifying square roots.

For further exploration and practice on simplifying the square root of 175, consider the following resources:

• : Provides a comprehensive explanation and examples on how to simplify square roots, including 175.
• : Offers detailed lessons and practice problems on simplifying radicals, useful for reinforcing your understanding.
• : Includes instructional videos and interactive exercises to help master simplifying radicals, including square roots.
• : Features step-by-step explanations and practice problems specifically focusing on simplifying square roots.