Square Root of 1.98

The square root is the inverse of the square of the number. 1.98 is not a perfect square. The square root of 1.98 is expressed in both radical and exponential form. In the radical form, it is expressed as √1.98, whereas (1.98)^(1/2) in the exponential form. √1.98 ≈ 1.407, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Since 1.98 is not a whole number, it cannot be broken down into prime factors like integers. Therefore, calculating 1.98 using prime factorization is not possible in the conventional sense. Instead, we use other methods such as long division or approximation.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. Since 1.98 is a decimal, we consider 198 after removing the decimal.

Step 2: Find n whose square is closest to 1. We use n = 1 because 1 × 1 is less than or equal to 1. The quotient is 1, and after subtracting 1 from 1, the remainder is 0.

Step 3: Bring down 98, making it the new dividend. Add the previous divisor 1 to itself, obtaining 2 as the new partial divisor.

Step 4: Find a digit x such that 2x × x is less than or equal to 98. It results in x = 4, since 24 × 4 = 96.

Step 5: Subtract 96 from 98, resulting in a remainder of 2, with a quotient of 1.4.

Step 6: Since the dividend is less than the divisor, add a decimal point and continue with zeroes. Bring down two zeros, making 200.

Step 7: Find a new digit that completes the divisor and multiply to get close to 200. The next digit is 1, making the quotient 1.41.

Step 8: Continue these steps to get a more accurate result.

The square root of 1.98 is approximately 1.407.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1.98 using the approximation method.

Step 1: Find the closest perfect squares around √1.98. The closest are 1 and 4, as √1 = 1 and √4 = 2. So, √1.98 falls between 1 and 2.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). (1.98 - 1) / (4 - 1) = 0.98 / 3 = 0.3267.

Step 3: Add this value to the smaller square root value: 1 + 0.3267 ≈ 1.3267. Adjusting for more precision through further approximation, we find √1.98 ≈ 1.407.

• Square root: A square root is the inverse of a square. Example: 1.4² ≈ 1.96, and the inverse of the square is the square root that is √1.96 ≈ 1.4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero, and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that is more prominent due to its uses in the real world. This is known as the principal square root.
   
• Decimals: If a number has a whole number and a fraction in a single number, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A method used to find the square root of numbers, especially non-perfect squares, by breaking down the number through systematic division.