How to Simplify Square Root of 72: A Comprehensive Guide

The square root of 72 can be simplified by expressing it in its simplest radical form. Here are the detailed steps to simplify √72:

Steps to Simplify √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 the list of factors:
   • 1, 4, 9, 36
3. Divide 72 by the largest perfect square:
   • \(72 / 36 = 2\)
4. Calculate the square root of the largest perfect square:
   • \(\sqrt{36} = 6\)
5. Combine the results to get the simplest form:
   • \(\sqrt{72} = 6\sqrt{2}\)

Square Root of 72 in Different Forms

Here are some different representations of the square root of 72:

• Radical form: \(\sqrt{72}\)
• Simplified radical form: \(6\sqrt{2}\)
• Exponential form: \(72^{1/2} = 6 \times 2^{1/2}\)
• Decimal form: \(\approx 8.48528137423857\)

Methods to Find the Square Root of 72

There are several methods to find the square root of 72, including:

• Repeated subtraction method
• Prime factorization method
• Long division method

Prime Factorization Method

Using prime factorization, we can express 72 as:

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

The square root of 72 can then be simplified as:

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

Long Division Method

1. Pair the digits of 72 from right to left.
2. Find a number whose square is less than or equal to 72 (which is 8).
3. Subtract and bring down pairs of zeros, continuing the process.
4. After several steps, the quotient obtained will be close to the value of \(\sqrt{72}\).

Using this method, we find:

\(\sqrt{72} \approx 8.48528137423857\)

Summary

The simplified form of \(\sqrt{72}\) is \(6\sqrt{2}\). This form is useful for exact calculations, while the decimal form (approximately 8.485) is helpful for practical approximations.

Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. The symbol for a square root is '√'. For instance, √9 equals 3 because 3 × 3 = 9. This process is the inverse of squaring a number.

Understanding square roots is crucial for various mathematical applications, including solving quadratic equations, working with exponents, and simplifying expressions. They are essential in fields such as engineering, physics, and computer science.

To calculate a square root, one can use methods such as prime factorization, the long division method, or estimation. Simplifying square roots involves breaking down a number into its prime factors to find the simplest radical form. This knowledge helps in handling more complex mathematical problems efficiently.

In this guide, we'll explore the square root of 72, demonstrating how to simplify it using different methods, and understand its various forms and representations.

The square root of 72, denoted as √72, is a value that, when multiplied by itself, gives the number 72. Square roots can be simplified to make calculations easier and to better understand their properties. Let's explore the square root of 72 in various forms.

First, we will look at the approximate decimal value of √72:

√72 ≈ 8.485

This value is obtained using a calculator, but it's important to understand the simplification process to express √72 in its simplest form.

Now, let's consider the prime factorization method to simplify √72:

• Start with the prime factorization of 72: 72 = 2 × 2 × 2 × 3 × 3
• Group the prime factors into pairs: (2 × 2) and (3 × 3) with a remaining 2
• Take one factor from each pair outside the square root: 2 × 3
• So, √72 = √(2 × 2 × 2 × 3 × 3) = 6√2

Therefore, the simplified radical form of √72 is 6√2.

To summarize:

• Decimal form: √72 ≈ 8.485
• Prime factorization: 72 = 2 × 2 × 2 × 3 × 3
• Simplified radical form: √72 = 6√2

Understanding these forms helps in different mathematical contexts, such as solving equations and comparing magnitudes.

Simplifying square roots involves reducing the radicand (the number inside the square root) to its simplest form. The goal is to find an equivalent expression that is easier to work with. To understand the simplification process, it's essential to know a few key concepts and steps:

Key Concepts:

• Radical Sign (√): The symbol used to denote the square root.
• Radicand: The number under the radical sign.
• Perfect Square: A number that has an integer as its square root (e.g., 4, 9, 16).
• Simplest Radical Form: An expression where the radicand has no perfect square factors other than 1.

Step-by-Step Process to Simplify Square Roots:

1. Factorize the Radicand: Break down the number under the square root into its prime factors.
2. Identify Perfect Squares: Look for pairs of prime factors, as these can be simplified further.
3. Simplify the Radical: Move the perfect square factors outside the radical, leaving the remaining factors inside.

Let's apply these steps to simplify the square root of 72:

1. Factorize 72: The prime factorization of 72 is 2 × 2 × 2 × 3 × 3.
2. Identify Perfect Squares: From the prime factors, we have (2 × 2) and (3 × 3), which are perfect squares.
3. Simplify: Take the square root of each perfect square:
   • √(2 × 2) = 2
   • √(3 × 3) = 3
   Combine these results outside the radical:

   √72 = √(2 × 2 × 2 × 3 × 3) = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2

The simplified form of √72 is 6√2, which means that 72 can be expressed as the product of 6 and the square root of 2.

Simplifying the square root of 72 involves breaking down the number into its prime factors and then simplifying the radical expression. Follow these detailed steps to simplify √72:

1. Prime Factorization:

   First, find the prime factors of 72. The prime factorization of 72 is:

   
   $$
   72
   =
   2
   ×
   2
   ×
   2
   ×
   3
   ×
   3
   

2. Group the Factors:

   Group the prime factors into pairs:

   
   $$
   72
   =
   (
   2
   ×
   2
   )
   ×
   2
   ×
   (
   3
   ×
   3
   )
   

3. Take the Square Root:

   Take the square root of each pair of factors and multiply them together:

   
   $$
   
   (
   2
   ×
   2
   )
   ×
   2
   ×
   (
   3
   ×
   3
   )
   
   =
   2
   ×
   3
   ×
   
   2
   
   

4. Combine the Results:

   Finally, combine the simplified factors:

   
   $$
   √72
   =
   6
   ×
   
   2
   
   =
   6
   
   2
   
   

Therefore, the simplest radical form of √72 is 6√2. This process involves factorizing the number, grouping factors into pairs, taking the square root of each pair, and combining the results to get the simplified form.

There are several methods to simplify square roots, each suited to different types of numbers and situations:

1. Prime Factorization Method: This method involves breaking down the number under the square root into its prime factors.
2. Estimation and Approximation: Useful for larger numbers or when exact simplification is not necessary, this method involves finding an approximate value close to the square root.
3. Long Division Method: Particularly useful for numbers that are not perfect squares, this method employs a process similar to long division to find a decimal approximation or simplified form.
4. Rationalizing Denominators: When dealing with square roots in fractions, this method involves multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the square root in the denominator.

Choosing the appropriate method depends on the complexity of the number and the desired form of the simplified square root.

The prime factorization method involves breaking down the number under the square root into its prime factors and then simplifying it. Here is the step-by-step process to simplify √72 using prime factorization:

1. Factorize 72 into Prime Factors:

   First, we find the prime factors of 72.

   72 = 2 × 2 × 2 × 3 × 3

2. Pair the Prime Factors:

   Next, we group the prime factors into pairs.

   72 = (2 × 2) × (3 × 3) × 2

3. Take One Factor from Each Pair Out of the Square Root:

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

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

4. Simplify:

   Multiply the factors taken out of the square root.

   2 × 3 = 6

   So, √72 simplifies to:

   √72 = 6√2

Therefore, the simplified form of √72 is 6√2.

The long division method is a systematic way to find the square root of a number. Here is a step-by-step guide to simplify √72 using this method:

1. Start by grouping the digits of the number in pairs from right to left. For 72, we have the pairs: (72).

2. Find the largest number whose square is less than or equal to the first group (72). In this case, 8 × 8 = 64, which is less than 72. Hence, the divisor and the first digit of the quotient is 8.

3. Subtract the square of the divisor from the first group: 72 - 64 = 8. Bring down a pair of zeros to the right of the remainder to get 800.

4. Double the quotient obtained so far (which is 8) and write it as 16. This is the new divisor. Find a digit to append to the new divisor and also to the quotient such that the new product is less than or equal to 800.

5. In this case, 164 × 4 = 656. Subtract 656 from 800 to get the new remainder, which is 144. Bring down another pair of zeros to get 14400.

6. Repeat the process: Double the quotient (84) to get 168 and find an appropriate digit to append. Here, 1688 × 8 = 13504, which is less than 14400.

7. Continue this process to get more precise digits after the decimal. The more pairs of zeros you bring down and divide, the more precise your square root will be.

Using the long division method, we find that the square root of 72 is approximately 8.485281.

This method ensures an accurate and detailed approach to finding square roots, making it a reliable technique for both simple and complex calculations.

The square root of 72 can be represented in various forms, each useful in different contexts. Here are the primary representations of √72:

Radical Form

The radical form is the most common way to express the square root:

\(\sqrt{72}\)

Simplified Radical Form

Simplifying the square root involves breaking it down into its prime factors and extracting any perfect squares. Here are the steps:

1. Factorize 72 into its prime factors: \(72 = 2^3 \times 3^2\).
2. Identify the perfect square factors: \(3^2 = 9\).
3. Simplify by taking the square root of the perfect square factor and placing it outside the radical:

\(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\)

Exponential Form

The square root can also be represented using exponents:

\(72^{\frac{1}{2}} = (36 \times 2)^{\frac{1}{2}} = 6 \times 2^{\frac{1}{2}}\)

Decimal Form

The decimal form provides an approximate numerical value of the square root:

\(\sqrt{72} \approx 8.48528\)

Each of these forms is useful depending on the situation. For exact values, the simplified radical form \(6\sqrt{2}\) is often preferred, while the decimal form is useful for practical calculations.

The square root of 72 can be simplified into its radical form by using the method of prime factorization. Here's how to simplify √72 step-by-step:

1. Factorize 72: First, find the prime factors of 72. The prime factorization of 72 is:
   • 72 = 2 × 2 × 2 × 3 × 3
2. Group the prime factors: Group the prime factors into pairs of the same number:
   • (2 × 2) × (3 × 3) × 2
3. Apply the square root: Take the square root of each pair of numbers. Each pair simplifies to a single number outside the radical:
   • √(2 × 2) = 2
   • √(3 × 3) = 3
4. Combine the results: Multiply the numbers outside the radical and place any remaining factors inside the radical:
   • 2 × 3 × √2 = 6√2

Therefore, the simplest radical form of the square root of 72 is:

√72 = 6√2

This simplified form indicates that √72 is equivalent to 6 times the square root of 2.

Using this simplified radical form can make it easier to work with square roots in mathematical expressions and equations.

To simplify the square root of 72, we need to express it in its simplest radical form. Here are the steps to achieve that:

1. Identify the factors of 72:
   • 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
2. Identify the largest perfect square factor of 72:
   • The perfect squares from the factors are 1, 4, 9, and 36.
   • The largest perfect square factor of 72 is 36.
3. Rewrite 72 as a product of 36 and another number:
   • \( 72 = 36 \times 2 \)
4. Use the property of square roots:
   • \( \sqrt{72} = \sqrt{36 \times 2} \)
   • \( \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} \)
5. Calculate the square root of 36:
   • \( \sqrt{36} = 6 \)
6. Combine the results:
   • \( \sqrt{72} = 6 \sqrt{2} \)

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

To express the square root of 72 in exponential form, we need to understand how exponents work. In general, any number can be represented as a base raised to an exponent. For square roots, the exponent is expressed as a fraction.

The square root of 72 can be written in exponential form as follows:

1. First, write the square root as a radical expression: \( \sqrt{72} \).
2. Then, convert the radical expression to an exponential form. The square root of a number is the same as raising that number to the power of \( \frac{1}{2} \).
3. Thus, \( \sqrt{72} \) can be written as \( 72^{\frac{1}{2}} \).

To simplify the exponential form, follow these steps:

1. Factorize the number 72 into its prime factors: \( 72 = 2^3 \times 3^2 \).
2. Apply the exponent \( \frac{1}{2} \) to each prime factor: \( (2^3 \times 3^2)^{\frac{1}{2}} \).
3. Use the property of exponents \( (a \times b)^n = a^n \times b^n \) to distribute the exponent: \( 2^{3 \times \frac{1}{2}} \times 3^{2 \times \frac{1}{2}} \).
4. Simplify the exponents: \( 2^{1.5} \times 3^1 = 2^{3/2} \times 3 \).

So, the simplified exponential form of the square root of 72 is:

\( 2^{3/2} \times 3 \).

This form is useful in many mathematical contexts, particularly in algebra and calculus, where operations involving exponents are common.

To express the square root of 72 in decimal form, we first need to understand its approximate value when evaluated numerically.

The square root of 72, denoted as \( \sqrt{72} \), can be calculated using a calculator or by manual computation. The result is approximately:

\[ \sqrt{72} \approx 8.4852813742386 \]

This value is the non-repeating, non-terminating decimal representation of \( \sqrt{72} \). It is a useful form when an exact decimal value is needed for practical applications, such as in measurements or when precision is required in calculations.

To summarize:

• The square root of 72 in its exact simplified radical form is \( 6\sqrt{2} \).
• In decimal form, the square root of 72 is approximately 8.4852813742386.

When using the decimal form, remember that it is an approximation and the digits continue infinitely without repeating. For most practical purposes, rounding to a desired number of decimal places (e.g., 8.49) is acceptable.

To better understand the simplification of the square root of 72, let's go through some examples and exercises.

Example 1: Simplify √72

1. First, find the prime factorization of 72: \( 72 = 2^3 \times 3^2 \).
2. Group the prime factors into pairs: \( \sqrt{72} = \sqrt{(2^2 \times 3^2) \times 2} \).
3. Take the square root of each pair: \( \sqrt{72} = 6\sqrt{2} \).

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

Example 2: Verify √72 in Decimal Form

1. Calculate the decimal form of \( 6\sqrt{2} \).
2. \( 6\sqrt{2} \approx 6 \times 1.414 = 8.485 \).

This verifies that \( \sqrt{72} \approx 8.485 \).

Exercise 1: Simplify the Square Root of 50

Follow the steps below:

1. Find the prime factorization of 50: \( 50 = 2 \times 5^2 \).
2. Group the prime factors: \( \sqrt{50} = \sqrt{(5^2) \times 2} \).
3. Take the square root of each pair: \( \sqrt{50} = 5\sqrt{2} \).

Thus, the simplified form of √50 is \( 5\sqrt{2} \).

Exercise 2: Simplify the Square Root of 98

Follow the steps below:

1. Find the prime factorization of 98: \( 98 = 2 \times 7^2 \).
2. Group the prime factors: \( \sqrt{98} = \sqrt{(7^2) \times 2} \).
3. Take the square root of each pair: \( \sqrt{98} = 7\sqrt{2} \).

Thus, the simplified form of √98 is \( 7\sqrt{2} \).

Exercise 3: Compare the Simplified Forms

Compare the simplified forms of √72, √50, and √98:

• \( \sqrt{72} = 6\sqrt{2} \)
• \( \sqrt{50} = 5\sqrt{2} \)
• \( \sqrt{98} = 7\sqrt{2} \)

Notice that all three square roots share the same radical part \( \sqrt{2} \), but differ by their coefficients.

Practice more with different numbers to get a better grasp of simplifying square roots!

Here are some commonly asked questions about square roots, including their properties and how to simplify them:

• What is a square root?

  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 \(3 \times 3 = 9\).

• How do you simplify the square root of a number?

  To simplify the square root of a number, you need to find the largest perfect square factor of the number and use the property that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). For instance, to simplify \(\sqrt{72}\), we recognize that 36 is the largest perfect square factor of 72.

  Therefore, \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6 \sqrt{2}\).

• What is the simplified radical form of \(\sqrt{72}\)?

  The simplified radical form of \(\sqrt{72}\) is \(6\sqrt{2}\).

• What is the decimal form of \(\sqrt{72}\)?

  The decimal form of \(\sqrt{72}\) is approximately 8.4853.

• What is the exponential form of \(\sqrt{72}\)?

  The exponential form of \(\sqrt{72}\) is \(72^{\frac{1}{2}}\) or \(6 \times 2^{\frac{1}{2}}\).

• Why is it important to simplify square roots?

  Simplifying square roots makes it easier to work with them in mathematical expressions and equations. It allows for a more concise and manageable form.

• What are some common mistakes when simplifying square roots?

  Common mistakes include not identifying the largest perfect square factor, incorrect multiplication or division, and forgetting to simplify the radical completely.

• How do you handle square roots of negative numbers?

  Square roots of negative numbers involve imaginary numbers. The square root of -1 is represented as \(i\), so the square root of a negative number like -72 is written as \(6i\sqrt{2}\).

Simplifying the square root of 72 can be approached in multiple ways, providing a thorough understanding of mathematical principles. Through this guide, we've explored several methods to achieve the simplified forms, such as:

• Prime Factorization
• Long Division
• Expressing in Different Forms (Radical, Simplified Radical, Exponential, Decimal)

Key takeaways include:

1. Prime Factorization: By breaking down 72 into its prime factors, we identified that 72 = 2^3 * 3^2. This allowed us to simplify √72 to 6√2.
2. Long Division: This method provided a step-by-step process to achieve a decimal approximation of √72, which is approximately 8.4853.
3. Various Representations: Understanding that √72 can be expressed in different forms (such as 6√2 in simplified radical form, and 8.4853 in decimal form) allows for flexibility in mathematical problem-solving.

Mastering the simplification of square roots not only enhances your mathematical skills but also prepares you for more advanced concepts. We hope this comprehensive guide has been beneficial and encourages further exploration of square roots and their applications.