Simplify 75 Square Root: Easy Steps to Master It

To simplify the square root of 75, we need to factorize 75 into its prime factors and identify any perfect squares.

Step-by-Step Simplification

1. Factorize 75:
   • 75 = 3 × 25
   • 25 = 5 × 5
   • Thus, 75 = 3 × 5 × 5
2. Identify the perfect squares:
   • 25 is a perfect square (5 × 5)
3. Rewrite the square root:
   • \(\sqrt{75} = \sqrt{3 \times 5^2}\)
4. Simplify by taking the square root of the perfect square:
   • \(\sqrt{75} = \sqrt{3 \times 5^2} = 5 \sqrt{3}\)

Final Result

The simplified form of \(\sqrt{75}\) is \(5 \sqrt{3}\).

Simplifying square roots involves expressing the square root in its simplest radical form. This process is essential in various mathematical applications and can make solving equations and understanding concepts easier. The goal is to break down the number inside the radical into its prime factors and then simplify it by identifying perfect squares.

Here is a step-by-step process to simplify square roots:

1. Identify the number inside the square root. For example, √75.
2. Find the prime factors of the number. For 75, the prime factors are 3, 5, and 5.
3. Group the prime factors into pairs. In this case, we have a pair of 5s.
4. Take the square root of each pair. The square root of 5^2 is 5.
5. Multiply the results from step 4 by any remaining factors. Here, √75 simplifies to 5√3.

This method ensures the square root is simplified to its lowest terms, making it more manageable in mathematical expressions.

The 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 \(5 \times 5 = 25\). This is denoted as \( \sqrt{25} = 5 \).

Perfect squares are numbers that are the squares of integers. In other words, a perfect square is a number that can be expressed as the product of an integer with itself. Some common perfect squares include:

• \(1 = 1 \times 1\)
• \(4 = 2 \times 2\)
• \(9 = 3 \times 3\)
• \(16 = 4 \times 4\)
• \(25 = 5 \times 5\)
• \(36 = 6 \times 6\)
• \(49 = 7 \times 7\)
• \(64 = 8 \times 8\)
• \(81 = 9 \times 9\)
• \(100 = 10 \times 10\)

Understanding perfect squares is essential when simplifying square roots because they help identify factors that can be simplified. When simplifying square roots, we look for perfect square factors of the number under the square root.

For example, to simplify \( \sqrt{75} \), we can use the perfect square factors of 75. Notice that:

• \(75 = 25 \times 3\)
• Since 25 is a perfect square, we can rewrite \( \sqrt{75} \) as \( \sqrt{25 \times 3} \)

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

\( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \)

Thus, the simplified form of \( \sqrt{75} \) is \( 5\sqrt{3} \). By recognizing and utilizing perfect squares, we can simplify square roots efficiently.

Simplifying the square root of 75 involves breaking down the number into its prime factors and identifying perfect squares. Here is a detailed, step-by-step guide:

1. List Factors of 75:

   First, identify all the factors of 75:

   • 1, 3, 5, 15, 25, 75

2. Identify Perfect Squares:

   From the list of factors, find the perfect squares:

   • 1, 25

3. Divide by the Largest Perfect Square:

   Divide 75 by the largest perfect square identified, which is 25:

   \[ \frac{75}{25} = 3 \]

4. Calculate the Square Root:

   Calculate the square root of the largest perfect square:

   \[ \sqrt{25} = 5 \]

5. Simplify the Square Root:

   Combine the results to express the square root of 75 in its simplest form:

   \[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \]

Thus, the simplified form of the square root of 75 is \( 5\sqrt{3} \).

The prime factorization method is a systematic approach to simplifying square roots. This method involves breaking down the number under the square root into its prime factors and then simplifying based on those factors.

Let's simplify the square root of 75 using the prime factorization method:

1. First, find the prime factors of 75.
   • 75 is divisible by 3: \(75 \div 3 = 25\)
   • Next, find the prime factors of 25:
     • 25 is divisible by 5: \(25 \div 5 = 5\)
     • 5 is divisible by 5: \(5 \div 5 = 1\)
2. Thus, the prime factors of 75 are: \(75 = 3 \times 5 \times 5\).
3. We can rewrite the square root of 75 using these prime factors: \[ \sqrt{75} = \sqrt{3 \times 5 \times 5} \]
4. Group the pairs of prime factors under the square root: \[ \sqrt{75} = \sqrt{3 \times 5^2} \]
5. Take the square root of the perfect square (5²) out of the radical: \[ \sqrt{75} = 5 \sqrt{3} \]

Therefore, the simplified form of \(\sqrt{75}\) is \(5 \sqrt{3}\).

Identifying perfect squares is a crucial step in simplifying square roots. Perfect squares are numbers that can be expressed as the square of an integer. For example, \(1, 4, 9, 16,\) and \(25\) are all perfect squares because they can be written as \(1^2, 2^2, 3^2, 4^2,\) and \(5^2\) respectively.

When simplifying the square root of a number like 75, we look for perfect square factors of 75. The steps to identify and use these perfect squares are as follows:

1. List the factors of the number:

   First, list all the factors of 75. These are 1, 3, 5, 15, 25, and 75.

2. Identify the perfect squares:

   From the list of factors, identify the perfect squares. In this case, the perfect square factors of 75 are 1 and 25.

3. Choose the largest perfect square:

   Select the largest perfect square factor, which is 25 in this instance.

4. Rewrite the square root:

   Express the square root of 75 using the perfect square factor:

   \[\sqrt{75} = \sqrt{25 \times 3}\]

5. Simplify the expression:

   Since 25 is a perfect square, we can simplify \(\sqrt{25}\) to 5. Therefore, the expression becomes:

   \[\sqrt{75} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}\]

By identifying and using perfect squares, we have simplified the square root of 75 to \(5\sqrt{3}\).

The simplification technique for square roots involves breaking down the number inside the square root into its prime factors and then simplifying the expression by extracting perfect squares. Here's a detailed step-by-step process to simplify the square root of 75:

1. Identify the prime factors of the number 75:

   \[
   75 = 3 \times 5^2
   \]

2. Express the square root of 75 using its prime factors:

   \[
   \sqrt{75} = \sqrt{3 \times 5^2}
   \]

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

   \[
   \sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2}
   \]

4. Simplify the square root of the perfect square:

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

5. Combine the simplified terms:

   \[
   \sqrt{75} = 5 \sqrt{3}
   \]

Therefore, the simplified form of \(\sqrt{75}\) is \(5\sqrt{3}\).

This method can be applied to any number by identifying and extracting perfect squares, leading to a simpler form of the square root.

To simplify the square root of 75, we start by breaking down 75 into its prime factors:

Next, identify pairs of identical prime factors to simplify:

• √(3 × 5 × 5)
• = √(3 × 5^2)
• = 5√3

Therefore, the simplified form of √75 is 5√3.

When simplifying the square root of 75, there are several common mistakes to be aware of:

1. Incorrect prime factorization of 75 can lead to errors in identifying perfect squares.
2. Forgetting to simplify the square root by identifying and factoring out perfect squares from under the radical.
3. Mistaking the square root of a product for the product of square roots.
4. Not verifying the simplified form by squaring it to ensure it equals the original number.

Let's practice simplifying the square root of 75 with the following examples:

1. Example 1:

   √75

   = √(3 × 25)

   = √(3 × 5 × 5)

   = √(3 × 5^2)

   = 5√3

   Therefore, √75 = 5√3.

2. Example 2:

   √75

   = √(15 × 5)

   = √(3 × 5 × 5)

   = √(3 × 5^2)

   = 5√3

   Thus, √75 simplifies to 5√3.

3. Practice Problem:

   Simplify √75 using the steps provided.

Simplified square roots, such as √75 = 5√3, are useful in various applications:

• Engineering calculations where simplified forms are easier to work with.
• Physics equations involving distances, velocities, and accelerations.
• Geometry problems requiring precise measurements and calculations.
• Financial modeling for estimating growth rates and compound interest.
• Computer graphics for scaling and transformations.

Simplifying the square root of 75 involves breaking down the number into its prime factors and identifying perfect squares to simplify. Through this process, we have learned:

• The prime factorization of 75 is 3 × 5 × 5.
• Identifying perfect squares helps in simplifying square roots efficiently.
• The simplified form of √75 is 5√3.

Understanding how to simplify square roots not only enhances problem-solving skills but also finds practical applications in various fields like mathematics, engineering, and physics.