Square Root 98 Simplified: A Comprehensive Guide

The square root of 98 can be simplified to its simplest radical form. Here's a detailed explanation of the process:

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For 98, we need to find a number \( q \) such that:

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

Prime Factorization Method

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

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

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

Long Division Method

Another method to find the square root is the long division method:

1. Group the digits of 98 in pairs from right to left.
2. Find the largest number whose square is less than or equal to 98, which is 9. Place it above the radical symbol.
3. Subtract the square of this number (81) from 98, giving a remainder of 17.
4. Double the quotient and use it as the new divisor. Continue the process to get a more precise value.

Using this method, we approximate \(\sqrt{98} \approx 9.899\).

Decimal Form

The decimal form of the square root of 98 is approximately:

\[
\sqrt{98} \approx 9.8994949366117
\]

Is 98 a Perfect Square?

No, 98 is not a perfect square because it cannot be expressed as the square of an integer.

Rational or Irrational?

The square root of 98 is an irrational number because it cannot be expressed as a simple fraction.

Summary

In summary, the square root of 98 is not a perfect square and is irrational. The simplified radical form is \( 7\sqrt{2} \), and its approximate decimal form is 9.899.

Square roots are mathematical operations that allow us to find a number which, when multiplied by itself, results in the original number. This operation is denoted by the radical symbol (√). The square root of a number \( x \) is a value \( y \) such that \( y^2 = x \). In this section, we will explore the concept of square roots, including their properties, how to calculate them, and their applications.

• The square root of a number \( x \) is a value \( y \) where \( y \times y = x \).
• Square roots can be either positive or negative, but the principal square root is the non-negative value.
• The square root of perfect squares (e.g., 1, 4, 9, 16) are integers.
• Non-perfect squares have irrational square roots, which cannot be expressed as a simple fraction.

Calculating the square root involves various methods such as prime factorization, long division, or using a calculator. For example, the square root of 98 can be simplified by expressing it in its prime factors and simplifying the radical expression:

1. Find the prime factors of 98: \( 98 = 2 \times 7 \times 7 \).
2. Group the factors in pairs: \( 98 = (7 \times 7) \times 2 \).
3. Take one factor from each pair out of the radical: \( \sqrt{98} = 7\sqrt{2} \).

Thus, the square root of 98 simplifies to \( 7\sqrt{2} \). Understanding square roots is fundamental in algebra and various mathematical applications, providing a basis for solving quadratic equations, analyzing geometric shapes, and more.

The square root of 98, represented as \( \sqrt{98} \), is the number which, when multiplied by itself, equals 98. In simplified radical form, the square root of 98 is expressed as \( 7\sqrt{2} \). This is derived through the process of prime factorization.

To simplify the square root of 98:

1. List the prime factors of 98: 2 and 49 (where 49 is \( 7^2 \)).
2. Rewrite 98 as \( 2 \times 7^2 \).
3. Apply the square root to the expression: \( \sqrt{98} = \sqrt{2 \times 7^2} \).
4. Simplify by separating the factors: \( \sqrt{2 \times 7^2} = 7\sqrt{2} \).

This simplified form \( 7\sqrt{2} \) indicates that 98 is not a perfect square, as its square root is an irrational number, approximately equal to 9.899 in decimal form.

To simplify the square root of 98, we use the method of prime factorization. This process involves breaking down the number 98 into its prime factors. Here are the steps to simplify √98:

1. Find the prime factors of 98. The prime factorization of 98 is:

   98 = 2 × 7 × 7 = 2 × 7²

2. Rewrite the square root of 98 using its prime factors:

   √98 = √(2 × 7²)

3. Separate the factors under the radical:

   √98 = √(7² × 2)

4. Take the square root of the perfect square (7²) out of the radical:

   √98 = 7√2

Therefore, the simplified form of the square root of 98 is 7√2.

This method shows that by breaking down the number into its prime factors, we can simplify the square root effectively.

The square root of 98 can be expressed in both simplified radical form and decimal form. In its simplified radical form, the square root of 98 is expressed as \(7\sqrt{2}\). However, when converted to decimal form, the value is approximately:

\(\sqrt{98} \approx 9.8994949366117\)

To understand this conversion in more detail, let’s break it down:

• The number 98 is not a perfect square, which means its square root will be an irrational number with a non-repeating, non-terminating decimal expansion.
• Using a calculator or computer, we can find that \(\sqrt{98} \approx 9.8995\) when rounded to four decimal places.

Here are some rounded forms of the square root of 98 for various decimal places:

• To the nearest tenth: \(\sqrt{98} \approx 9.9\)
• To the nearest hundredth: \(\sqrt{98} \approx 9.90\)
• To the nearest thousandth: \(\sqrt{98} \approx 9.899\)

If you need to represent the square root of 98 in different ways, you can use the following forms:

• Fractional Exponent: \(\sqrt{98} = 98^{1/2}\)
• Approximate Fraction: While \(\sqrt{98}\) cannot be exactly expressed as a fraction due to its irrational nature, it can be approximated. For example, \(\sqrt{98} \approx \frac{990}{100}\) or \(9 \frac{9}{10}\).

Using tools like calculators, you can quickly determine the decimal form by entering 98 and pressing the square root button (√). On most standard calculators and software like Excel, the SQRT function will give you the result \( \sqrt{98} \approx 9.8994949366117 \).

This decimal representation is particularly useful in practical applications where an approximate value is sufficient, such as in measurements and engineering calculations.

The square root of 98 is an irrational number. To understand why, let's first define what makes a number rational or irrational:

• Rational Number: A number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, where \(p\) and \(q\) are integers and \(q \neq 0\).
• Irrational Number: A number that cannot be expressed as a simple fraction - it's decimal form is non-repeating and non-terminating.

Now, let's consider the square root of 98:

When we simplify the square root of 98 using the prime factorization method, we get:

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

Here, \(7\) is a rational number and \(\sqrt{2}\) is known to be an irrational number. The product of a rational number (7) and an irrational number (\(\sqrt{2}\)) is always irrational. Thus, \(7\sqrt{2}\) is an irrational number.

To verify, let's approximate the value of \(\sqrt{98}\):

\[
\sqrt{98} \approx 9.899494937
\]

The decimal form of \(\sqrt{98}\) is non-repeating and non-terminating, which further confirms that it is an irrational number.

Therefore, we conclude that the square root of 98 is irrational.

Calculating the square root of 98 using a calculator is straightforward. Here are the detailed steps:

1. Turn on your calculator.
2. Locate the square root (√) button on your calculator. This is often labeled as "√" or "sqrt."
3. Press the square root button (√).
4. Enter the number 98.
5. Press the equals (=) button to calculate the result.
6. The display will show the square root of 98, which is approximately 9.899494937.

For a more detailed understanding, let's represent this mathematically:

$$

98

=
9.899494937

Using a calculator provides a quick and accurate way to find the square root of any number, including 98. This method is especially useful for large numbers or when a precise decimal value is needed.

There are several methods to find the square root of 98. Below are three commonly used methods:

Prime Factorization Method

1. Find the prime factors of 98:

   98 can be written as \( 2 \times 7^2 \).

2. Group the prime factors into pairs:

   \( 98 = 2 \times (7^2) \).

3. Take the square root of each pair:

   \( \sqrt{98} = \sqrt{2 \times 7^2} = 7\sqrt{2} \).

Long Division Method

1. Start by pairing the digits of 98 from right to left (98).
2. Find the largest number whose square is less than or equal to the first pair (9):

   9 × 9 = 81, so the first digit is 9.

3. Subtract 81 from 98 to get the remainder 17.
4. Bring down pairs of zeros and continue the process to find the decimal value.
5. For more precision, continue the division process:

   The result is approximately 9.899.

Repeated Subtraction Method

1. Start with the number 98.
2. Subtract successive odd numbers starting from 1 (i.e., 1, 3, 5, 7, ...) until you reach zero or a negative number.
3. Count the number of subtractions performed:

   98 - 1 = 97

   97 - 3 = 94

   94 - 5 = 89

   89 - 7 = 82

   82 - 9 = 73

   73 - 11 = 62

   62 - 13 = 49

   49 - 15 = 34

   34 - 17 = 17

   17 - 19 = -2

4. The number of steps is approximately the square root of 98, giving a rough estimate.

Each of these methods provides a way to find the square root of 98, with varying levels of precision and complexity.

To understand how to simplify square roots, let's look at a few examples:

• Square Root of 98:

  The prime factorization of 98 is 2 and 49 (since 98 = 2 × 7 × 7). Therefore, √98 can be simplified as:

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

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

• Square Root of 50:

  The prime factorization of 50 is 2 and 25 (since 50 = 2 × 5 × 5). Therefore, √50 can be simplified as:

  \[
  \sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}
  \]

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

• Square Root of 72:

  The prime factorization of 72 is 2 and 36 (since 72 = 2 × 36). Therefore, √72 can be simplified as:

  \[
  \sqrt{72} = \sqrt{2^3 \times 3^2} = 6\sqrt{2}
  \]

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

• Square Root of 45:

  The prime factorization of 45 is 3 and 15 (since 45 = 3 × 3 × 5). Therefore, √45 can be simplified as:

  \[
  \sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5}
  \]

  So, the simplified form of √45 is \( 3\sqrt{5} \).

• Square Root of 20:

  The prime factorization of 20 is 2 and 10 (since 20 = 2 × 2 × 5). Therefore, √20 can be simplified as:

  \[
  \sqrt{20} = \sqrt{2^2 \times 5} = 2\sqrt{5}
  \]

  So, the simplified form of √20 is \( 2\sqrt{5} \).

Understanding how to simplify the square root of 98 can be highly beneficial in both academic and practical applications. By applying the prime factorization method, we found that the square root of 98 can be simplified to 7√2. This simplification process not only aids in better comprehension of square roots but also enhances problem-solving skills in mathematics.

We explored various methods to find the square root, such as the prime factorization method, the long division method, and the repeated subtraction method. Each method offers a unique approach, allowing us to choose the most suitable one based on the context of the problem.

Through this exploration, we discovered that 98 is not a perfect square, and its square root is an irrational number. This means that the square root of 98 cannot be expressed as a simple fraction, and its decimal form is non-repeating and non-terminating.

Using a calculator, we can approximate the square root of 98 to a decimal value, which is approximately 9.899. This approximation is useful for practical purposes where an exact value is not necessary.

In conclusion, the process of simplifying and understanding the square root of 98 provides a solid foundation for tackling more complex mathematical problems. It highlights the importance of prime factorization and other methods in simplifying square roots, and emphasizes the distinction between rational and irrational numbers. With these insights, we are better equipped to handle mathematical challenges involving square roots.