Square Root of 2/9

The square root is the inverse of the square of a number. 2/9 is not a perfect square. The square root of 2/9 is expressed in both radical and exponential form. In the radical form, it is expressed as √(2/9), whereas (2/9)^(1/2) in exponential form. √(2/9) = √2/√9 = √2/3, 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. Now let us look at how 2/9 is broken down into its prime factors:

Step 1: Finding the prime factors of 2 and 9. Breaking it down, we get 2 as 2 and 9 as 3 x 3: 2 and 3^2.

Step 2: Now we found out the prime factors of 2/9. The second step is to make pairs of those prime factors. Since 2/9 is not a perfect square, calculating the square root of 2/9 using prime factorization involves taking the square root of the numerator and the denominator separately.

The long division method is particularly used for non-perfect square numbers. In this method, we can check by dividing 2 by 9 first, and then finding the square root. Let us now learn how to find the square root using the long division method, step by step:

Step 1: Divide 2 by 9 to get 0.222.

Step 2: Find the square root of 0.222... using the long division method.

Step 3: Group the digits in pairs from the decimal point, i.e., 0.22 | 20.

Step 4: Find the largest number whose square is less than or equal to 2.00, which is 1. The divisor becomes 1.

Step 5: Subtract 1 from 2.00 to get 1.00, then bring down the next pair to get 100.

Step 6: Double the current divisor (1) to get 2, and find a digit n such that 2n × n is less than or equal to 100. The digit is 4 because 24 × 4 = 96.

Step 7: Subtract 96 from 100 to get 4, then bring down the next pair to get 400.

Step 8: Continue this process until you get the desired precision.

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 2/9 using the approximation method.

Step 1: Approximate the square root of 2/9 by estimating the square roots of the numerator and denominator. √2 is approximately 1.414 and √9 is exactly 3.

Step 2: Divide the approximate square root of the numerator by the square root of the denominator: 1.414/3 ≈ 0.471

Step 3: This approximate value is the square root of 2/9, i.e., 0.471.

• Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and the inverse operation is the square root: √16 = 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.
   
• Fraction: A fraction represents a part of a whole and is expressed as a ratio of two integers, such as 1/2, 3/4, etc.
   
• Perimeter: The perimeter is the total length of the boundary of a two-dimensional shape. For a rectangle, it is calculated as 2 × (length + width).
   
• Approximation: Approximation is the process of finding a value that is close enough to the correct answer, usually within a specified range.