Square Root of 1989

The square root is the inverse of the square of the number. 1989 is not a perfect square. The square root of 1989 is expressed in both radical and exponential form. In the radical form, it is expressed as √1989, whereas in the exponential form, it is expressed as (1989)^(1/2). √1989 ≈ 44.5925, 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, for non-perfect square numbers, 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. Now let us look at how 1989 is broken down into its prime factors.

Step 1: Finding the prime factors of 1989

Breaking it down, we get 3 x 3 x 3 x 3 x 11 x 2: 3^3 x 11 x 2

Step 2: Now we found the prime factors of 1989. Since 1989 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √1989 using prime factorization is impractical.

The long division method is particularly used for non-perfect square numbers. In this method, we determine 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, we group the numbers from right to left. In the case of 1989, we group it as 89 and 19.

Step 2: Now we need to find n whose square is closest to 19. The closest perfect square is 16 (4^2), so we choose n as 4. Now the quotient is 4.

Step 3: Subtract 16 from 19 to get 3, then bring down 89 to get a new dividend of 389.

Step 4: Double the quotient (4) to get 8, which will be part of our new divisor.

Step 5: We need to find a digit x such that 8x * x is less than or equal to 389. Let's try x = 4, giving 84 * 4 = 336.

Step 6: Subtract 336 from 389 to get 53. The quotient is now 44.

Step 7: Add a decimal point to the quotient and bring down two zeros to make the new dividend 5300.

Step 8: Double the quotient part before the decimal (44) to get 88.

Step 9: Find a digit y such that 88y * y is less than or equal to 5300. Let's try y = 6, giving 886 * 6 = 5316.

Step 10: Subtract 5316 from 5300 to get -16. Since we need to refine our guess, let's try y = 5, giving 885 * 5 = 4425.

Step 11: Subtract 4425 from 5300 to get 875. The quotient is now 44.5. Continue these steps until the desired precision is achieved.

The square root of √1989 ≈ 44.5925.

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 1989 using the approximation method.

Step 1: Identify the closest perfect squares to √1989. The smallest perfect square less than 1989 is 1936 (44^2) and the largest perfect square more than 1989 is 2025 (45^2). Therefore, √1989 falls between 44 and 45.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Using the formula: (1989 - 1936) / (2025 - 1936) = 53 / 89 ≈ 0.5955 Adding this value to 44 gives us approximately 44.5955, so the square root of 1989 is approximately 44.5955.

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16 and the inverse of the square is the square root that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction p/q, where q is not zero and p and q are integers.
   
• Approximation: A method used to find an estimated value that is close to the actual value, often used when dealing with irrational numbers.
   
• Decimal: A number that contains a whole number and a fractional part separated by a decimal point, for example: 44.59.
   
• Radical: The symbol used to denote the square root (√) or nth root of a number.