Simplify Square Root of 84: A Step-by-Step Guide

To simplify the square root of 84, we need to find the prime factorization of 84 and identify any perfect squares.

Step-by-Step Simplification

1. Find the prime factors of 84:
   • 84 ÷ 2 = 42
   • 42 ÷ 2 = 21
   • 21 ÷ 3 = 7
   • 7 is a prime number
2. Write the prime factorization of 84:

   84 = 2 × 2 × 3 × 7

3. Group the prime factors into pairs:

   \(84 = 2^2 \times 3 \times 7\)

4. Take the square root of each pair and simplify:

   \(\sqrt{84} = \sqrt{2^2 \times 3 \times 7} = 2 \sqrt{21}\)

Final Simplified Form

The simplified form of the square root of 84 is \(2 \sqrt{21}\).

Square roots are a fundamental concept in mathematics, representing a number that, when multiplied by itself, gives the original number. Simplifying square roots involves breaking down the number under the root to its simplest form. This process can make calculations easier and results more understandable.

To simplify square roots, we use the prime factorization method. Here’s a step-by-step approach:

1. Find the prime factors of the number under the square root.
2. Group the prime factors into pairs.
3. Move each pair of prime factors outside the square root.
4. Multiply the factors outside the square root together.
5. Leave any unpaired prime factors inside the square root.

Let’s apply this method to the square root of 84 to understand the simplification process better.

Square roots are an essential part of mathematics, representing the value that, when multiplied by itself, equals the original number. For example, the square root of 25 is 5 because \(5 \times 5 = 25\). The square root symbol is denoted by \( \sqrt{} \).

When working with square roots, there are a few key concepts to understand:

1. Perfect Squares: Numbers like 1, 4, 9, 16, and 25 are perfect squares because they are the product of an integer multiplied by itself.
2. Non-Perfect Squares: Numbers like 2, 3, 5, and 84 are not perfect squares because their square roots are not whole numbers. These roots can be simplified but will still involve a square root symbol.
3. Simplification: Simplifying a square root means expressing it in the simplest radical form. This involves finding the prime factorization of the number and grouping the factors.
4. Prime Factorization: This is the process of breaking down a number into its basic prime factors. For example, the prime factorization of 84 is \(2^2 \times 3 \times 7\).

By understanding these concepts, you can simplify square roots effectively. Simplification often involves identifying pairs of prime factors and taking them outside the square root, as pairs represent perfect squares.

The prime factorization method is a systematic way to simplify square roots by breaking down the number under the square root into its prime factors. Here's a detailed step-by-step guide to using this method:

1. Identify the Number:

   Start with the number you want to simplify, which in this case is 84.

2. Find Prime Factors:

   Determine the prime factors of 84. Prime factors are the prime numbers that multiply together to give the original number. For 84, the prime factorization is:

   • 84 ÷ 2 = 42
   • 42 ÷ 2 = 21
   • 21 ÷ 3 = 7
   • 7 is a prime number

   So, the prime factorization of 84 is \(2 \times 2 \times 3 \times 7\), or \(2^2 \times 3 \times 7\).

3. Group the Prime Factors:

   Group the prime factors into pairs where possible:

   \(84 = 2^2 \times 3 \times 7\)

4. Simplify the Square Root:

   For each pair of prime factors, take one factor out of the square root:

   \(\sqrt{84} = \sqrt{2^2 \times 3 \times 7} = 2 \sqrt{21}\)

5. Final Result:

   The simplified form of \(\sqrt{84}\) is \(2 \sqrt{21}\).

Using the prime factorization method helps simplify square roots by reducing them to their simplest form, making calculations easier and results more precise.

Simplifying the square root of 84 involves breaking it down into its simplest form using prime factorization. Follow these detailed steps:

1. Find the Prime Factors:

   Determine the prime factors of 84. Start by dividing 84 by the smallest prime number, which is 2:

   • 84 ÷ 2 = 42
   • 42 ÷ 2 = 21
   • 21 ÷ 3 = 7
   • 7 is a prime number

   So, the prime factorization of 84 is \(2^2 \times 3 \times 7\).

2. Group the Prime Factors:

   Group the prime factors into pairs:

   \(84 = 2^2 \times 3 \times 7\)

3. Simplify the Square Root:

   Take one number from each pair of prime factors out of the square root. For each pair of 2s, take a single 2 out:

   \(\sqrt{84} = \sqrt{2^2 \times 3 \times 7} = 2 \sqrt{21}\)

4. Verify the Simplification:

   Ensure that the simplification is correct by checking if \(2 \sqrt{21}\) squared gives back 84:

   \((2 \sqrt{21})^2 = 2^2 \times (\sqrt{21})^2 = 4 \times 21 = 84\)

Thus, the simplified form of \(\sqrt{84}\) is \(2 \sqrt{21}\). This method makes it easier to handle square roots by reducing them to their simplest form.

To simplify the square root of 84, we first need to break down the number 84 into its prime factors. Here’s a detailed step-by-step process:

1. Identify the Number:

   We start with the number 84, which we aim to simplify under the square root.

2. Divide by the Smallest Prime Number:

   Begin by dividing 84 by the smallest prime number, which is 2:

   • 84 ÷ 2 = 42
3. Continue Dividing by 2:

   Continue dividing the result by 2 until it is no longer divisible by 2:

   • 42 ÷ 2 = 21
4. Move to the Next Prime Number:

   Since 21 is not divisible by 2, move to the next smallest prime number, which is 3:

   • 21 ÷ 3 = 7
5. Prime Number Check:

   7 is a prime number and cannot be divided further by any prime number other than itself.

6. Prime Factorization Result:

   Thus, the prime factorization of 84 is:

   \(84 = 2 \times 2 \times 3 \times 7\) or \(2^2 \times 3 \times 7\).

Breaking down 84 into its prime factors helps in simplifying the square root. These factors are used to group and simplify under the square root sign.

After breaking down the number 84 into its prime factors, the next step in simplifying the square root is to group these prime factors. Here’s how you do it:

1. Prime Factorization:

   The prime factorization of 84 is \(2^2 \times 3 \times 7\).

2. Identify Pairs of Prime Factors:

   Look for pairs of the same prime factors. In this case, the number 2 appears twice:

   • \(2^2\) represents a pair of 2s.
3. Group the Pairs:

   Group the pairs together. Each pair will be taken out of the square root:

   • \(\sqrt{2^2 \times 3 \times 7} = \sqrt{(2 \times 2) \times 3 \times 7}\)
4. Simplify the Square Root:

   For each pair of prime factors, take one factor out of the square root. In this example, take one 2 out of the square root:

   • \(\sqrt{(2 \times 2) \times 3 \times 7} = 2 \sqrt{3 \times 7}\)
   • Simplify the remaining factors under the square root:
   • \(2 \sqrt{3 \times 7} = 2 \sqrt{21}\)
5. Final Simplified Form:

   The final simplified form of the square root of 84 is \(2 \sqrt{21}\).

By grouping the prime factors, you can simplify the square root effectively, making calculations easier and more understandable.

To simplify the square root of 84, we need to identify and extract perfect square factors from the number. This process involves the following steps:

1. Start with the prime factorization of 84:

   The prime factorization of 84 is:

   • 84 = 2 × 2 × 3 × 7
2. Group the prime factors into pairs of the same number:

   From the prime factors, we can see that there is one pair of 2s.

3. Extract the pairs as perfect squares:

   The pair of 2s can be grouped together as 2², which is a perfect square.

   This can be written as:

   • 2² = 4
4. Simplify the square root by extracting the perfect squares:

   Now, rewrite the square root of 84 as a product of square roots:

   • √84 = √(2² × 3 × 7)

   Since √(2²) = 2, we can extract the 2 from under the square root:

   • √84 = 2√(3 × 7)
5. Combine the remaining factors:

   Multiply the remaining factors under the square root:

   • √84 = 2√21

Therefore, the simplified form of √84 is:

2√21

To simplify the square root of 84, we need to factor the number into its prime factors and identify any perfect squares.

1. First, perform the prime factorization of 84:
   • 84 can be factored into 2 × 42.
   • 42 can be factored into 2 × 21.
   • 21 can be factored into 3 × 7.
2. This gives us the prime factors of 84:

   84 = 2 × 2 × 3 × 7

3. Next, group the prime factors into pairs where possible:

   84 = (2 × 2) × 3 × 7

4. Identify the perfect squares in these groups. In this case, (2 × 2) is a perfect square:

   √84 = √((2 × 2) × 3 × 7)

5. Take the square root of the perfect square (2 × 2):

   √(2 × 2) = 2

6. Rewrite the expression by taking the square root of the perfect square outside the radical:

   √84 = 2√(3 × 7)

7. Finally, simplify the expression under the radical:

   √84 = 2√21

Thus, the simplified form of √84 is 2√21.

In decimal form, this can be approximated as:

2√21 ≈ 9.165

There are several alternative methods for simplifying square roots. Here, we will explore a few of them to provide a comprehensive understanding of how to simplify √84.

• Prime Factorization Method

This method involves breaking down the number into its prime factors and then simplifying.

1. Find the prime factors of 84: 84 = 2 × 2 × 3 × 7.
2. Group the prime factors into pairs: (2 × 2) × 3 × 7.
3. Take the square root of each pair and simplify: √(2 × 2) × √(3 × 7) = 2√21.

Thus, the simplified form is 2√21.

• Division Method

This method is a systematic approach similar to long division, often used to find more precise decimal values.

1. Estimate a number that, when squared, is close to 84. We use 9 because 9² = 81.
2. Divide 84 by 9 to get a quotient (approximately 9.3333).
3. Average 9 and 9.3333 to refine the estimate. The result is around 9.1666.
4. Continue this process until the desired accuracy is achieved.

For practical purposes, √84 is approximately 9.165.

• Using Approximation Techniques

This method involves using known squares and estimating between them.

1. Identify the perfect squares closest to 84: 81 (9²) and 100 (10²).
2. Since 84 is between these two squares, √84 is between 9 and 10.
3. Estimate more precisely by considering smaller intervals. For instance, since 84 is closer to 81, try values closer to 9.

Using this technique, we find that √84 is approximately 9.165.

By understanding and applying these methods, you can simplify square roots effectively and choose the most appropriate method depending on the context.

When simplifying square roots, it's essential to be aware of common mistakes that can lead to incorrect results. Here are some pitfalls to watch out for and how to avoid them:

• Incorrect Factorization: One common mistake is not correctly factoring the number under the square root. For example, failing to recognize that 84 can be factored into 4 and 21. Ensure you list all factors and identify the largest perfect square.
• Ignoring Perfect Squares: Always identify and use the largest perfect square factor. For √84, the largest perfect square is 4, leading to 2√21. Skipping this step can lead to an unsimplified form.
• Improper Grouping: When using the prime factorization method, it's crucial to correctly group the factors into pairs. For example, √84 = √(4×21) = √(2²×21) = 2√21.
• Forgetting to Simplify Completely: Ensure that the square root is in its simplest form. For instance, √84 simplifies to 2√21. Not completing the simplification can result in an incomplete answer.
• Mistakes in Multiplication: When simplifying, ensure that you multiply the factors correctly. Any error in basic arithmetic can lead to incorrect results.
• Misapplication of Properties: Remember that √(a × b) = √a × √b. Incorrectly applying this property can result in incorrect simplifications. For example, √84 = √(4×21) = 2√21, not √(2²×21).

By avoiding these common mistakes, you can ensure that you correctly simplify square roots and achieve accurate results.

The square root of 84, simplified as 2√21, has various practical applications across different fields. Understanding and utilizing this simplified form can enhance problem-solving skills and provide valuable insights in several areas:

• Geometry and Architecture: In geometry, the square root of 84 can be used to calculate dimensions and areas. For instance, if a square has an area of 84 square units, finding the length of one side involves calculating the square root of 84. This is crucial for precise measurements in architectural designs.
• Algebraic Problems: The simplified form 2√21 is often used in algebra for solving quadratic equations, inequalities, and simplifying expressions. This serves as a foundation for more complex mathematical concepts and enhances algebraic manipulation skills.
• Physics and Engineering: In physics, the square root of 84 might be used in equations involving distances, velocities, or energy levels. For example, it could be part of a formula to calculate the magnitude of a vector or the resultant force in a system. In engineering, it can assist in designing components and analyzing stress and strain.
• Statistics and Probability: The square root function is essential in statistics, particularly in calculating standard deviation and variance. Understanding how to simplify square roots like √84 is crucial for data analysis and interpreting statistical results.
• Financial Estimates: In finance, square roots are used in various models and calculations, such as risk assessment and investment growth predictions. The ability to simplify square roots aids in these complex calculations, providing more accurate and manageable results.

The versatility of the square root of 84 demonstrates its importance in both academic and practical applications. Mastery of such mathematical concepts enhances overall problem-solving abilities and contributes to a deeper understanding of various scientific and mathematical disciplines.

In conclusion, simplifying the square root of 84 involves breaking down the number into its prime factors and extracting any perfect squares. The number 84 can be factored as:

\[ 84 = 2^2 \times 21 \]

By taking the square root of the perfect square, we get:

\[ \sqrt{84} = \sqrt{2^2 \times 21} = 2\sqrt{21} \]

This gives us the simplified form of the square root of 84 as \( 2\sqrt{21} \).

Understanding how to simplify square roots is not only useful in mathematical computations but also has practical applications in various fields such as geometry, physics, and engineering. Simplified square roots make it easier to perform calculations and understand the relationships between different quantities.

To summarize the process:

• Break down the number into its prime factors.
• Identify and extract any perfect squares.
• Express the simplified form using the remaining factors under the square root.

The simplified form of the square root of 84 is \( 2\sqrt{21} \), which is approximately equal to 9.165 when expressed as a decimal.

Through this comprehensive guide, we have explored various methods to simplify square roots, understood the importance of prime factorization, and learned about the practical applications of simplified square roots in real-world scenarios.