Simplify the Square Root of 72: A Step-by-Step Guide

In this guide, we will break down the process of simplifying the square root of 72 in a clear and concise manner. Let's dive in!

Step-by-Step Process

1. Identify the prime factors of 72.
2. Pair the prime factors.
3. Simplify the expression using the pairs.

Prime Factorization

The first step in simplifying \(\sqrt{72}\) is to find the prime factors of 72. Here's the prime factorization:

72 = 2 \times 2 \times 2 \times 3 \times 3

Grouping the Factors

Next, we group the prime factors into pairs:

72 = (2 \times 2) \times (3 \times 3) \times 2

Simplifying the Expression

Now, we take the square root of each pair:

\(\sqrt{72} = \sqrt{(2 \times 2) \times (3 \times 3) \times 2}\)

Simplifying further, we get:

\(\sqrt{72} = 2 \times 3 \times \sqrt{2}\)

Therefore, the simplified form of \(\sqrt{72}\) is:

\(\sqrt{72} = 6\sqrt{2}\)

Conclusion

We have successfully simplified \(\sqrt{72}\) to 6\sqrt{2}. By following these steps, you can simplify any square root by finding its prime factors and grouping them into pairs. Happy simplifying!

The process of simplifying square roots involves breaking down a number into its simplest radical form. This often makes calculations easier and expressions more manageable. Simplifying the square root of 72, for example, can be done by factoring the number into smaller components and applying mathematical rules. Let's explore how this process works step by step.

1. Identify the factors of the number under the square root. For 72, the factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

2. Find the largest perfect square among the factors. For 72, the largest perfect square is 36.

3. Rewrite the original number as a product of the perfect square and another factor: \(72 = 36 \times 2\).

4. Apply the square root to both factors: \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}\).

5. Simplify the square root of the perfect square: \(\sqrt{36} = 6\).

6. Combine the results to get the simplified form: \(\sqrt{72} = 6\sqrt{2}\).

This method helps in simplifying square roots by making use of perfect squares, which are easier to manage. Understanding and practicing this process can be extremely useful in various mathematical applications, including algebra and geometry.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). The square root symbol is \( \sqrt{} \), and the number inside the symbol is called the radicand.

To express the concept mathematically:

If \( \sqrt{a} = b \), then \( b^2 = a \).

For example:

• \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \)
• \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \)

Square roots can also be represented in different forms:

• Radical Form: \( \sqrt{72} \)
• Exponential Form: \( 72^{1/2} \)
• Decimal Form: approximately 8.4853

There are different methods to simplify square roots, such as:

• Prime Factorization
• Long Division

Let's explore these methods briefly:

Prime Factorization Method

This method involves breaking down the radicand into its prime factors. For example, to simplify \( \sqrt{72} \), we find the prime factors of 72:

1. 72 can be factored into \( 2 \times 36 \)
2. 36 can be factored into \( 6 \times 6 \)
3. 6 can be factored into \( 2 \times 3 \)

So, \( 72 = 2 \times 2 \times 2 \times 3 \times 3 \). Grouping the prime factors in pairs, we get \( (2 \times 2) \times (3 \times 3) \times 2 \). Therefore:

\( \sqrt{72} = \sqrt{(2^2 \times 3^2 \times 2)} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} \).

Long Division Method

The long division method is used for finding square roots by manually dividing and finding the root step-by-step. This method is particularly useful for more complex numbers where factoring is not straightforward.

Understanding square roots is essential for simplifying radical expressions and solving various mathematical problems. The square root of a non-perfect square is an irrational number, meaning its decimal form is non-repeating and non-terminating.

For example, the square root of 72, which is \( \sqrt{72} \), can be simplified to \( 6\sqrt{2} \), and its approximate decimal value is 8.4853.

By grasping these fundamental concepts, you can tackle more complex mathematical problems involving square roots with confidence.

The prime factorization method is a systematic approach to simplify square roots by breaking down the number into its prime factors. Here is a step-by-step guide to simplifying the square root of 72 using the prime factorization method:

1. First, find the prime factors of 72. Start by dividing 72 by the smallest prime number, which is 2:

   \( 72 \div 2 = 36 \)

2. Continue dividing by 2 until you can no longer divide evenly:

   \( 36 \div 2 = 18 \)

   \( 18 \div 2 = 9 \)

3. Now, divide 9 by the next smallest prime number, which is 3:

   \( 9 \div 3 = 3 \)

   \( 3 \div 3 = 1 \)

The prime factors of 72 are \( 2 \times 2 \times 2 \times 3 \times 3 \), which can be written as \( 2^3 \times 3^2 \).

Next, group the prime factors into pairs:

\( 72 = (2 \times 2) \times (3 \times 3) \times 2 \)

Take the square root of each pair, and move the pair outside the radical:

\( \sqrt{72} = \sqrt{(2 \times 2) \times (3 \times 3) \times 2} \)

\( \sqrt{72} = 2 \times 3 \times \sqrt{2} \)

Simplify the expression:

\( \sqrt{72} = 6\sqrt{2} \)

Therefore, the simplified form of \( \sqrt{72} \) is \( 6\sqrt{2} \).

This method is reliable and can be used to simplify any square root by breaking it down into its prime factors and grouping them into pairs.

Simplifying the square root of 72 involves breaking it down into its prime factors and using those factors to simplify the expression. Follow these steps to simplify √72:

1. Factorize 72:

   First, determine the prime factors of 72. The factors are:

   72 = 2 × 2 × 2 × 3 × 3

2. Group the Factors:

   Next, group the factors into pairs of equal factors:

   (2 × 2) × (3 × 3) × 2

3. Take the Square Root of Each Pair:

   The square root of each pair of equal factors is a single number. So, we have:

   √(2 × 2) = 2

   √(3 × 3) = 3

4. Simplify the Expression:

   Combine the results of the square roots taken from each pair. Any factor that does not form a pair remains under the square root:

   √72 = √(2 × 2 × 2 × 3 × 3) = 2 × 3 × √2

5. Final Simplified Form:

   Thus, the simplified form of the square root of 72 is:

   √72 = 6√2

This step-by-step process helps to break down the problem into manageable parts, making it easier to understand and solve. The final result shows that the square root of 72, simplified, is 6√2.

To simplify the square root of 72, we first need to break down the number 72 into its prime factors. Prime factorization is the process of expressing a number as the product of its prime factors. Here's a step-by-step guide:

1. Find the smallest prime number that divides 72: The smallest prime number is 2. Since 72 is even, it is divisible by 2.

   72 ÷ 2 = 36

2. Repeat the process with the quotient: The quotient is 36, which is also even and divisible by 2.

   36 ÷ 2 = 18

3. Continue dividing by 2: The quotient is 18, which is even and divisible by 2.

   18 ÷ 2 = 9

4. Divide by the next smallest prime number: The quotient is 9, which is divisible by 3 (the next smallest prime number).

   9 ÷ 3 = 3

5. Complete the factorization: The quotient is 3, which is divisible by itself (3).

   3 ÷ 3 = 1

After performing these steps, we can express 72 as the product of its prime factors:

\[72 = 2 \times 2 \times 2 \times 3 \times 3\]

Or, more compactly:

\[72 = 2^3 \times 3^2\]

This prime factorization is essential for simplifying the square root of 72, as it allows us to identify and extract perfect squares, leading to the simplest radical form.

Once we have identified the prime factors of 72, the next step is to group and pair these factors to simplify the square root. Let's revisit the prime factorization of 72:

\( 72 = 2^3 \times 3^2 \)

To simplify the square root, we need to find pairs of prime factors. Each pair of identical factors will come out of the square root as a single factor.

1. First, let's list the factors:

\( 2 \times 2 \times 2 \times 3 \times 3 \)

2. Next, group the factors into pairs:
• We have one pair of \( 2s \): \( 2 \times 2 \)
• And one pair of \( 3s \): \( 3 \times 3 \)
• We also have one \( 2 \) left unpaired.
3. Take one factor out of each pair:

\( \sqrt{72} = \sqrt{(2 \times 2) \times (3 \times 3) \times 2} = 2 \times 3 \times \sqrt{2} \)

4. Multiply the factors outside the radical:

\( 2 \times 3 = 6 \)

5. Combine the results:

\( \sqrt{72} = 6\sqrt{2} \)

Thus, by grouping and pairing the factors, we have simplified the square root of 72 to \( 6\sqrt{2} \).

Now that we have broken down the prime factors of 72 and grouped them, we can calculate the simplified square root. Here is a step-by-step guide:

1. We have identified that the prime factors of 72 are \(2 \times 2 \times 2 \times 3 \times 3\).

2. We grouped these factors into pairs of equal factors: \((2 \times 2) \times (3 \times 3) \times 2\).

3. Now, we take the square root of each pair of factors:

   • \(\sqrt{2 \times 2} = 2\)
   • \(\sqrt{3 \times 3} = 3\)

4. Since the last factor \(2\) does not form a pair, it remains under the square root sign.

5. We multiply the results of the square roots of the pairs together with the remaining factor under the square root:

   \(2 \times 3 \times \sqrt{2} = 6\sqrt{2}\)

6. Therefore, the simplified form of the square root of 72 is \(6\sqrt{2}\).

This simplified form is much easier to work with in mathematical calculations and provides a clear understanding of the value of the square root of 72.

Understanding how to simplify the square root of 72 is not just a mathematical exercise; it has practical applications in various fields such as engineering, architecture, and physics. Here are some examples that illustrate its usefulness:

• Geometry and Architecture: Calculating the diagonal length of square and rectangular spaces. For instance, the simplified square root of 72 can help architects determine the diagonal of a square room whose area is 72 square units.

  Consider a square room with an area of 72 square units. To find the length of the diagonal, you can use the simplified square root of 72. Since the area is \(72\) square units, each side of the square is \(\sqrt{72} = 6\sqrt{2}\). The diagonal of the square can be calculated using the Pythagorean theorem:

  \[
  \text{Diagonal} = \sqrt{(\text{side})^2 + (\text{side})^2} = \sqrt{(6\sqrt{2})^2 + (6\sqrt{2})^2} = \sqrt{72 + 72} = \sqrt{144} = 12
  \]

• Physics and Engineering: Simplifying square roots is essential for calculating distances, speeds, and other measurements in problems involving the Pythagorean theorem. Engineers might use the simplified form of \(\sqrt{72}\) to calculate the stress or force in structures.

  For example, in physics, calculating the resultant force when two forces of equal magnitude act at a right angle to each other involves the square root of 72. If each force has a magnitude of \(6\sqrt{2}\), the resultant force \(F\) is:

  \[
  F = \sqrt{(6\sqrt{2})^2 + (6\sqrt{2})^2} = \sqrt{72 + 72} = \sqrt{144} = 12
  \]

• Education and Problem Solving: Simplification techniques are crucial for students learning algebra and geometry, helping them solve equations and understand the properties of radicals.

  In educational settings, teachers use the square root of 72 to demonstrate the process of simplifying radicals. By breaking down 72 into its prime factors (2 x 2 x 2 x 3 x 3) and grouping them, students learn that the simplified form of \(\sqrt{72}\) is \(6\sqrt{2}\).

• Real-life Applications: From designing electronic components to calculating the area of land, the ability to simplify square roots like \(\sqrt{72}\) is widely applicable in day-to-day problem-solving scenarios.

  For instance, when determining the side length of a square plot of land with an area of 72 square units, knowing that \(\sqrt{72} = 6\sqrt{2}\) can help in planning and optimizing space usage.

When simplifying the square root of 72, certain pitfalls can hinder the accuracy of your solution. Being aware of these common mistakes can enhance your understanding and ensure a correct simplification process.

• Overlooking Prime Factorization:

  A common mistake is not fully breaking down 72 into its prime factors. Ensure you reach the most basic prime numbers for accurate simplification.

• Incorrect Grouping of Factors:

  Misgrouping the prime factors can lead to erroneous results. Remember that you need pairs of the same factor to simplify under the square root.

• Forgetting to Multiply Outside the Root:

  After extracting square roots from the prime factors, some forget to multiply these values outside the square root, which is crucial for reaching the final simplified form.

• Misplacing the Leftover Factor:

  It's easy to forget or incorrectly place the leftover prime factor that doesn’t form a pair. This factor must remain under the square root for correct simplification.

• Confusing Addition with Multiplication:

  Ensure that when you’re simplifying, you multiply the numbers outside the square root rather than adding them. This is a common error that changes the outcome significantly.

By avoiding these mistakes, you can simplify the square root of 72 accurately and efficiently, enhancing your mathematical skills and understanding.

• What is the simplified form of the square root of 72?

  The simplified form of the square root of 72 is \(6\sqrt{2}\). This is achieved by breaking down 72 into its prime factors and pairing them under the radical sign.

• How do you simplify the square root of 72 step by step?
  1. List the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
  2. Identify the perfect squares from these factors: 1, 4, 9, 36.
  3. Divide 72 by the largest perfect square: \(72 \div 36 = 2\).
  4. Take the square root of the largest perfect square: \(\sqrt{36} = 6\).
  5. Combine the results: \(6\sqrt{2}\).
• Can you explain why \(6\sqrt{2}\) is the simplest form?

  Yes, \(6\sqrt{2}\) is the simplest form because 36 is the largest perfect square factor of 72. When 72 is expressed as \(36 \times 2\), the square root of 36 is 6, leaving \(\sqrt{2}\) as the remaining factor under the radical.

• What are some practical applications of simplifying square roots?

  Simplifying square roots is useful in various fields such as engineering, physics, and computer science, where precise calculations involving roots are necessary. It helps in solving equations, optimizing algorithms, and in simplifying expressions for better readability and understanding.

• Are there any common mistakes to avoid when simplifying square roots?

  Yes, common mistakes include not identifying the largest perfect square factor, incorrectly pairing the factors, and not fully simplifying the radical expression. Ensuring each step is followed correctly will help avoid these errors.

In conclusion, simplifying the square root of 72 involves breaking down the number into its prime factors and pairing them under the radical sign. The result, \(6\sqrt{2}\), is the simplest form. Understanding this process enhances your mathematical skills and helps in various practical applications.

Here’s a summary of the steps involved in simplifying \(\sqrt{72}\):

1. List the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
2. Identify the perfect squares from these factors: 1, 4, 9, 36.
3. Divide 72 by the largest perfect square: \(72 \div 36 = 2\).
4. Take the square root of the largest perfect square: \(\sqrt{36} = 6\).
5. Combine the results: \(6\sqrt{2}\).

For further reading, consider exploring the following topics:

• : A comprehensive guide on simplifying various square roots using different methods.
• : An online tool for simplifying square roots and other mathematical expressions.
• : Detailed steps and explanations for simplifying the square root of 72.

By continuing to practice and explore these resources, you will strengthen your understanding of simplifying square roots and other mathematical concepts.