Simplify Square Root 75: A Step-by-Step Guide to Mastering Radicals

Simplifying square roots involves expressing the square root in its simplest radical form. Let's simplify \( \sqrt{75} \) step by step.

Step 1: Prime Factorization

First, find the prime factors of 75. The prime factorization of 75 is:

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

Step 2: Break Down the Square Root

Next, express the square root of 75 using its prime factors:

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

Step 3: Separate the Factors

Separate the factors inside the square root into pairs:

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

Step 4: Simplify the Square Root of the Perfect Square

Since the square root of \( 5^2 \) is 5, we can simplify this part:

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

Step 5: Combine the Results

Combine the simplified square root with the remaining factor:

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

Conclusion

Thus, the simplified form of \( \sqrt{75} \) is:

\[
\boxed{5\sqrt{3}}
\]

In this section, we will simplify the square root of 75 step-by-step. Simplifying radicals helps in making expressions easier to understand and work with. The square root of 75 can be simplified by identifying its factors and perfect squares.

1. List the factors of 75: \(1, 3, 5, 15, 25, 75\).
2. Identify the perfect squares from the list of factors: \(1, 25\).
3. Divide 75 by the largest perfect square you found: \(75 \div 25 = 3\).
4. Calculate the square root of the largest perfect square: \(\sqrt{25} = 5\).
5. Combine the results to get the simplest form: \(5\sqrt{3}\).

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

Square roots are a fundamental concept in mathematics, representing a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 × 3 = 9. The square root symbol is √, and the square root of a number \( x \) is written as \( \sqrt{x} \).

Understanding how to simplify square roots is crucial for solving various mathematical problems. Simplification involves expressing the square root in its simplest form. Let's delve into the steps to understand this better.

1. Identify Perfect Squares: Look for factors of the number under the square root that are perfect squares. Perfect squares are numbers like 1, 4, 9, 16, 25, etc., which have whole numbers as their square roots.
2. Rewrite the Square Root: Break down the number under the square root into its prime factors and group the factors into pairs of identical numbers. For example, to simplify \( \sqrt{75} \), we first find the factors of 75: 1, 3, 5, 15, 25, and 75. The perfect square factor here is 25.
3. Divide and Simplify: Divide the original number by the largest perfect square factor identified. For \( \sqrt{75} \), divide 75 by 25 to get 3. Thus, \( \sqrt{75} = \sqrt{25 \times 3} \).
4. Apply the Square Root to the Perfect Square: Take the square root of the perfect square. Since \( \sqrt{25} = 5 \), we can rewrite \( \sqrt{75} \) as \( 5\sqrt{3} \).

This step-by-step method helps in simplifying square roots efficiently. By breaking down the process, we ensure that the square root expression is in its simplest form, making it easier to work with in further calculations.

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. Simplifying the square root of a number often involves finding its prime factors. Here’s a step-by-step guide to prime factorizing the number 75:

1. First, identify the prime factors of 75. We can start by dividing 75 by the smallest prime number, which is 2. Since 75 is odd, it is not divisible by 2.
2. Next, try dividing by 3, the next smallest prime number. 75 divided by 3 equals 25.
3. Now we need to factor 25. Since 25 is a perfect square, it can be factored into \(5 \times 5\).
4. Therefore, the prime factorization of 75 is \(3 \times 5^2\).

Once the prime factorization is complete, we can use these factors to simplify the square root:

• Rewrite the square root of 75 using its prime factors: \(\sqrt{75} = \sqrt{3 \times 5^2}\).
• Apply the property of square roots that allows us to separate the factors: \(\sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2}\).
• Simplify the square root of the perfect square: \(\sqrt{5^2} = 5\).
• Thus, the simplified form of the square root of 75 is \(5\sqrt{3}\).

This process of prime factorization and simplifying square roots helps in understanding the structure of numbers and is a fundamental skill in algebra.

To simplify the square root of 75, follow these detailed steps:

1. Prime Factorization:

   Begin by finding the prime factors of 75. The number 75 can be factored into:

   • \(75 = 3 \times 5 \times 5\)

2. Rewrite the Square Root:

   Express the square root of 75 using its prime factors:

   \(\sqrt{75} = \sqrt{3 \times 5^2}\)

3. Separate the Factors:

   Separate the square root into the product of square roots:

   \(\sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2}\)

4. Simplify the Perfect Square:

   Simplify the square root of the perfect square (\(5^2\)):

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

5. Combine the Results:

   Combine the simplified terms to get the final simplified form:

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

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

Below are some example problems to help you practice simplifying the square root of 75 and similar expressions.

• Example 1: Simplify \( \sqrt{75} \)

  1. Factor 75 into its prime factors: \( 75 = 3 \times 5^2 \).
  2. Rewrite the square root as a product of square roots: \( \sqrt{75} = \sqrt{3 \times 5^2} \).
  3. Simplify the square root of the perfect square: \( \sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2} = \sqrt{3} \times 5 \).
  4. Thus, \( \sqrt{75} = 5\sqrt{3} \).

• Example 2: Simplify \( 2\sqrt{75} \)

  1. Use the simplification from Example 1: \( \sqrt{75} = 5\sqrt{3} \).
  2. Multiply by 2: \( 2\sqrt{75} = 2 \times 5\sqrt{3} = 10\sqrt{3} \).

• Example 3: Simplify \( \sqrt{50} \)

  1. Factor 50 into its prime factors: \( 50 = 2 \times 5^2 \).
  2. Rewrite the square root as a product of square roots: \( \sqrt{50} = \sqrt{2 \times 5^2} \).
  3. Simplify the square root of the perfect square: \( \sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2} = \sqrt{2} \times 5 \).
  4. Thus, \( \sqrt{50} = 5\sqrt{2} \).

• Example 4: Simplify \( 3\sqrt{45} \)

  1. Factor 45 into its prime factors: \( 45 = 3^2 \times 5 \).
  2. Rewrite the square root as a product of square roots: \( \sqrt{45} = \sqrt{3^2 \times 5} \).
  3. Simplify the square root of the perfect square: \( \sqrt{3^2 \times 5} = \sqrt{3^2} \times \sqrt{5} = 3\sqrt{5} \).
  4. Multiply by 3: \( 3\sqrt{45} = 3 \times 3\sqrt{5} = 9\sqrt{5} \).

• Example 5: Simplify \( \sqrt{300} \)

  1. Factor 300 into its prime factors: \( 300 = 3 \times 100 \).
  2. Rewrite the square root as a product of square roots: \( \sqrt{300} = \sqrt{3 \times 100} \).
  3. Simplify the square root of the perfect square: \( \sqrt{3 \times 100} = \sqrt{3} \times \sqrt{100} = \sqrt{3} \times 10 \).
  4. Thus, \( \sqrt{300} = 10\sqrt{3} \).

Practicing these examples can help you get better at simplifying square roots, especially when dealing with larger numbers or more complex expressions.

When simplifying the square root of 75, there are several common mistakes that students often make. By being aware of these pitfalls, you can avoid errors and simplify square roots more effectively.

• Forgetting to Check for Perfect Squares: Always check if the number inside the square root has perfect square factors. For instance, 75 can be factored into 25 and 3, where 25 is a perfect square.
• Incorrect Prime Factorization: Ensure that you correctly factorize the number into its prime factors. For 75, the prime factors are 3 and 5 (75 = 3 × 5 × 5).
• Rushing the Process: Simplification requires careful steps. Rushing through can lead to mistakes. Take your time to factorize and simplify step by step.
• Misinterpreting the Square Root Symbol: Remember that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). For example, \(\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}\).
• Neglecting to Simplify Completely: After factoring, ensure that you simplify the expression fully. For instance, from \(\sqrt{75} = 5\sqrt{3}\), check if further simplification is possible.

By avoiding these common mistakes, you can simplify square roots more accurately and efficiently.

Here are some common questions and answers related to simplifying the square root of 75.

• What is the simplified form of the square root of 75?

  The simplified form of \( \sqrt{75} \) is \( 5\sqrt{3} \). This is because 75 can be factored into 25 and 3, and the square root of 25 is 5.

• How do you simplify the square root of 75?
  1. Factor the number 75 into its prime factors: \( 75 = 3 \times 5^2 \).
  2. Rewrite the square root using these factors: \( \sqrt{75} = \sqrt{3 \times 5^2} \).
  3. Split the square root: \( \sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2} \).
  4. Simplify the square root of the perfect square: \( \sqrt{5^2} = 5 \).
  5. Combine the simplified terms: \( 5\sqrt{3} \).
• Why can't we simplify the square root of 75 further?

  After simplifying \( \sqrt{75} \) to \( 5\sqrt{3} \), the term \( \sqrt{3} \) cannot be simplified further because 3 is a prime number and not a perfect square.

• Can you provide an example of another square root simplification?

  Sure! Let's simplify the square root of 45.

  1. Factor 45 into its prime factors: \( 45 = 9 \times 5 \).
  2. Rewrite the square root: \( \sqrt{45} = \sqrt{9 \times 5} \).
  3. Split the square root: \( \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} \).
  4. Simplify the square root of the perfect square: \( \sqrt{9} = 3 \).
  5. Combine the simplified terms: \( 3\sqrt{5} \).
• What is the purpose of simplifying square roots?

  Simplifying square roots makes them easier to work with in mathematical expressions and equations. It helps in recognizing equivalent values and can make further calculations simpler and more efficient.

To further understand the process of simplifying the square root of 75, here are some practice problems. Try to solve them step-by-step and check your answers below.

• Problem 1: Simplify \(\sqrt{45}\)

  1. Find the largest perfect square factor of 45.
  2. Rewrite \(\sqrt{45}\) as a product of square roots.
  3. Simplify the square roots.

  Solution:

  1. The largest perfect square factor of 45 is 9.
  2. \(\sqrt{45} = \sqrt{9 \times 5}\)
  3. \(\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\)

• Problem 2: Simplify \(\sqrt{200}\)

  1. Find the largest perfect square factor of 200.
  2. Rewrite \(\sqrt{200}\) as a product of square roots.
  3. Simplify the square roots.

  Solution:

  1. The largest perfect square factor of 200 is 100.
  2. \(\sqrt{200} = \sqrt{100 \times 2}\)
  3. \(\sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}\)

• Problem 3: Simplify \(\sqrt{72}\)

  1. Find the largest perfect square factor of 72.
  2. Rewrite \(\sqrt{72}\) as a product of square roots.
  3. Simplify the square roots.

  Solution:

  1. The largest perfect square factor of 72 is 36.
  2. \(\sqrt{72} = \sqrt{36 \times 2}\)
  3. \(\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\)

• Problem 4: Simplify \(\sqrt{50}\)

  1. Find the largest perfect square factor of 50.
  2. Rewrite \(\sqrt{50}\) as a product of square roots.
  3. Simplify the square roots.

  Solution:

  1. The largest perfect square factor of 50 is 25.
  2. \(\sqrt{50} = \sqrt{25 \times 2}\)
  3. \(\sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)

• Problem 5: Simplify \(\sqrt{27}\)

  1. Find the largest perfect square factor of 27.
  2. Rewrite \(\sqrt{27}\) as a product of square roots.
  3. Simplify the square roots.

  Solution:

  1. The largest perfect square factor of 27 is 9.
  2. \(\sqrt{27} = \sqrt{9 \times 3}\)
  3. \(\sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\)

These practice problems should help reinforce your understanding of simplifying square roots. Remember, the key steps are to identify the largest perfect square factor and then simplify accordingly.