Square Root of 2.14

The square root is the inverse of the square of a number. 2.14 is not a perfect square. The square root of 2.14 is expressed in both radical and exponential form. In the radical form, it is expressed as √2.14, whereas \(2.14^{1/2}\) in the exponential form. √2.14 ≈ 1.46287, 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, methods like the long-division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation method

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 2.14 is a small number, we consider 214 as a whole.

Step 2: We need to find a number whose square is less than or equal to 2. The closest number is 1, as 1 × 1 = 1. Subtracting 1 from 2 gives a remainder of 1.

Step 3: Bring down 14 to make it 114. The new divisor is 2 (2 × 1).

Step 4: Find a number n such that 2n × n ≤ 114. Consider n as 5, since 25 × 5 = 125, which is more than 114. So, consider n as 4, where 24 × 4 = 96.

Step 5: Subtract 96 from 114 to get 18. The quotient is 1.4.

Step 6: Since 18 is less than the divisor, we add a decimal and bring down two zeros to make it 1800. Continue this process until you get the desired decimal places.

The square root of 2.14 is approximately 1.46287.

The approximation method is another way to find square roots. It is an easy method to approximate the square root of a given number. Let us learn how to find the square root of 2.14 using the approximation method.

Step 1: Identify the closest perfect squares around 2.14. The closest perfect squares are 1 (√1 = 1) and 4 (√4 = 2).

Step 2: 2.14 is closer to 1 than to 4. Calculate the approximate value by interpolation. If we assume the progression between 1 and 4 is linear, we can estimate the square root of 2.14 by finding its position between the two perfect squares. Using a linear approximation, the square root of 2.14 is approximately 1.46287.

• Square root: A square root is the inverse of squaring a number. Example: 4² = 16, and the inverse of the square is the square root, so √16 = 4.
   
• Irrational number: An irrational number cannot be written as a simple fraction; its decimal form is non-repeating and non-terminating.
   
• Rational number: A rational number can be expressed as a fraction where both the numerator and the denominator are integers.
   
• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point, such as 7.86.
   
• Linear interpolation: A method of estimating unknown values that lie between two known values, often used for approximation in mathematics.