98 Square Root Simplified: A Step-by-Step Guide to Understanding and Mastery

The process of simplifying the square root of a number involves expressing it in its simplest radical form. For the number 98, we can break it down into its prime factors and simplify accordingly.

Step-by-Step Simplification

1. Find the prime factorization of 98.
   • 98 is an even number, so it is divisible by 2:
   • 98 ÷ 2 = 49
   • 49 is not divisible by 2, but it is divisible by 7:
   • 49 ÷ 7 = 7
   • 7 is a prime number.
   • Therefore, the prime factorization of 98 is \( 2 \times 7^2 \).
2. Express the square root of 98 using its prime factors:

   \[ \sqrt{98} = \sqrt{2 \times 7^2} \]

3. Separate the factors inside the square root:

   \[ \sqrt{98} = \sqrt{2} \times \sqrt{7^2} \]

4. Simplify the square root of the perfect square:

   \[ \sqrt{7^2} = 7 \]

5. Combine the simplified parts:

   \[ \sqrt{98} = 7 \times \sqrt{2} \]

Conclusion

Therefore, the simplest form of the square root of 98 is:

\[ \sqrt{98} = 7 \sqrt{2} \]

Square roots are fundamental concepts in mathematics that help us understand the dimensions and properties of numbers and shapes. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\).

Square roots are represented using the radical symbol \(\sqrt{}\). For instance, the square root of 16 is written as \(\sqrt{16}\). When we talk about square roots, it's important to distinguish between perfect squares and non-perfect squares:

• Perfect Squares: Numbers like 1, 4, 9, 16, 25, etc., which have exact square roots.
• Non-Perfect Squares: Numbers like 2, 3, 5, 7, 10, etc., which do not have exact square roots and are often simplified into irrational numbers.

Understanding how to simplify square roots is crucial for various mathematical applications, including algebra, geometry, and calculus. Simplifying square roots involves breaking down a number into its prime factors and finding pairs of factors that can be simplified.

In this guide, we will focus on the square root of 98. Although 98 is not a perfect square, we can simplify \(\sqrt{98}\) by using the prime factorization method. This process will help us better understand the properties and applications of square roots in different mathematical contexts.

The square root of 98, denoted as \( \sqrt{98} \), is an irrational number. This means that it cannot be expressed as a simple fraction and its decimal representation is non-terminating and non-repeating. To understand \( \sqrt{98} \) better, let's explore its properties and the steps to simplify it.

Properties of the Square Root of 98

• Radical Form: \( \sqrt{98} \)
• Decimal Form: \( \sqrt{98} \approx 9.899 \)
• Exponential Form: \( 98^{1/2} \)
• Irrational Number: Since 98 is not a perfect square, \( \sqrt{98} \) is an irrational number.

Simplifying the Square Root of 98

To simplify \( \sqrt{98} \), we use the method of prime factorization.

1. First, find the prime factors of 98:
   • 98 can be factored into \( 2 \times 49 \)
   • 49 can be further factored into \( 7 \times 7 \)
2. So, the prime factorization of 98 is \( 2 \times 7^2 \).
3. Next, we apply the square root to these factors:
   • \( \sqrt{98} = \sqrt{2 \times 7^2} \)
   • Separate the perfect square factor: \( \sqrt{2} \times \sqrt{7^2} = \sqrt{2} \times 7 \)
   • Therefore, \( \sqrt{98} = 7\sqrt{2} \)

Verification of Simplified Form

To verify the simplified form, we can square \( 7\sqrt{2} \) and check if it equals 98:

\[
(7\sqrt{2})^2 = 7^2 \times (\sqrt{2})^2 = 49 \times 2 = 98
\]

Thus, our simplification is correct.

Decimal Approximation

Using a calculator, the decimal approximation of \( \sqrt{98} \) is:

\[
\sqrt{98} \approx 9.899
\]

In conclusion, \( \sqrt{98} \) simplifies to \( 7\sqrt{2} \) and approximately equals 9.899 in decimal form.

To simplify the square root of 98 using the prime factorization method, follow these steps:

1. List Factors: First, list the factors of 98. The factors are 1, 2, 7, 14, 49, and 98.
2. Identify Perfect Squares: From the list of factors, identify the perfect squares. The perfect squares in the list are 1 and 49.
3. Divide by Largest Perfect Square: Divide 98 by the largest perfect square from the previous step. Here, 49 is the largest perfect square.
   • 98 ÷ 49 = 2
4. Calculate Square Root: Calculate the square root of the perfect square.
   • √49 = 7
5. Simplify Radical Expression: Combine the results of the division and square root calculation to get the simplest form of the square root of 98.
   • √98 = √(49 × 2) = √49 × √2 = 7√2

Therefore, the simplified form of the square root of 98 is \(7\sqrt{2}\).

Let's summarize this process in a table:

To simplify the square root of 98, we will use the prime factorization method. Follow these steps to get the simplest form:

1. List the Factors:

   First, identify all the factors of 98. They are 1, 2, 7, 14, 49, and 98.

2. Find the Perfect Squares:

   Among these factors, the perfect squares are 1 and 49.

3. Divide by the Largest Perfect Square:

   Divide 98 by the largest perfect square from the list:

   \[ 98 \div 49 = 2 \]

4. Calculate the Square Root:

   Find the square root of the largest perfect square:

   \[ \sqrt{49} = 7 \]

5. Combine the Results:

   Combine the results from steps 3 and 4 to get the simplified form:

   \[ \sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2} \]

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

The approximate decimal form is:

\[ 7\sqrt{2} \approx 9.899 \]

To verify that the simplified form of the square root of 98 is correct, we need to follow a few steps:

1. Original Simplification:

   The simplified form of \(\sqrt{98}\) is \(7\sqrt{2}\).

2. Square the Simplified Form:

   We square \(7\sqrt{2}\) to see if it equals 98:

   \[
   (7\sqrt{2})^2 = 7^2 \times (\sqrt{2})^2 = 49 \times 2 = 98
   \]

3. Comparison:

   Since \(49 \times 2 = 98\), the simplified form \(7\sqrt{2}\) is verified as correct.

Let's break down the verification process in more detail:

1. First, recall the simplified form of \(\sqrt{98}\):

   \(\sqrt{98} = 7\sqrt{2}\)

2. Next, square the simplified form:

   \((7\sqrt{2})^2\)

3. Expand the squared term:

   \[
   (7\sqrt{2})^2 = 7^2 \times (\sqrt{2})^2
   \]

4. Calculate the squares of the individual terms:

   \[
   7^2 = 49 \quad \text{and} \quad (\sqrt{2})^2 = 2
   \]

5. Multiply the results:

   \[
   49 \times 2 = 98
   \]

6. Finally, since the original number 98 matches the result of our calculation, the simplified form \(7\sqrt{2}\) is confirmed as correct.

Simplified square roots, such as that of 98, find practical use in various mathematical and real-world contexts. Here are some applications:

1. Estimation: Simplified square roots help quickly estimate the magnitude of numbers, aiding in mental math and approximate calculations.
2. Geometry: In geometric calculations, simplified square roots are often used to find side lengths or areas of squares, rectangles, and circles.
3. Engineering: Engineers frequently encounter square roots in fields like structural design, where simplified forms are used for efficient material and structural calculations.
4. Financial Analysis: Financial analysts use square roots in risk assessment models and statistical calculations, where simplified forms aid in quicker analysis.
5. Physics: In physics, simplified square roots are utilized in equations involving velocity, acceleration, and energy calculations.

These applications demonstrate the practical utility of simplified square roots across diverse fields, showcasing their importance beyond theoretical mathematics.

When simplifying the square root of 98 or similar expressions, it's crucial to avoid these common errors:

• Incorrect Factorization: Mistakes in identifying and factoring the components of 98 can lead to incorrect simplification.
• Missing Simplification Steps: Skipping essential steps in the simplification process can result in an incomplete or inaccurate simplified form.
• Ignoring Rationalization: For expressions involving square roots with non-perfect squares, failure to rationalize the denominator can obscure the simplified form.
• Overlooking Common Factors: Not identifying and canceling out common factors during simplification may prevent achieving the simplest form.
• Incorrect Application of Rules: Applying square root rules incorrectly, such as distributing incorrectly across terms or misusing properties of square roots, can lead to errors.

By avoiding these mistakes, you ensure accurate and efficient simplification of square roots like 98, enhancing your mathematical proficiency and understanding.

Practice simplifying the square root of 98 with the following problems:

1. Simplify \( \sqrt{98} \).
2. Verify your result by multiplying the simplified form back to check if it equals 98.
3. Find another way to simplify \( \sqrt{98} \) using a different method or approach.
4. Explore applications of the simplified square root of 98 in real-world scenarios, such as estimating quantities or solving geometric problems.
5. Create your own practice problems involving square roots of similar numbers for further mastery.

Simplifying the square root of 98 involves understanding its prime factors and applying mathematical principles effectively. By mastering the process, you can:

• Enhance your problem-solving skills in mathematical contexts.
• Apply simplification techniques to various mathematical and practical scenarios.
• Gain confidence in handling square roots of numbers through practice and application.
• Avoid common mistakes by paying attention to factorization and simplification steps.
• Utilize simplified square roots for estimation and quick calculations in real-world situations.

Overall, simplifying the square root of 98 not only improves your mathematical proficiency but also equips you with valuable skills applicable across different disciplines and everyday challenges.