Square Root of 3.14

The square root is the inverse of the square of a number. 3.14 is not a perfect square. The square root of 3.14 is expressed in both radical and exponential form. In radical form, it is expressed as √3.14, whereas in exponential form it is expressed as (3.14)^(1/2). √3.14 ≈ 1.772, which is an irrational number because it cannot be expressed in the form 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:

• Long division method
• Approximation method

The long division method is particularly used for non-perfect square numbers. This method involves dividing the number in a manner similar to traditional division. 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 pair the numbers from the decimal point to the left and right. In the case of 3.14, we consider 3 and 14.

Step 2: Find a number whose square is less than or equal to 3. We use 1 because 1 × 1 = 1. The quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 14, making the new dividend 214. Double the quotient (1) and use it as the new divisor's first digit, giving us 2_.

Step 4: Find a digit 'n' such that 2n × n is less than or equal to 214. Try n = 7, because 27 × 7 = 189.

Step 5: Subtract 189 from 214, leaving a remainder of 25. The quotient is 1.7.

Step 6: Add a decimal point and bring down two zeros, making the new dividend 2500.

Step 7: Find a new divisor, 34 (since 2 × 17 = 34), and determine n such that 34n × n is less than or equal to 2500. Try n = 7, because 347 × 7 = 2429.

Step 8: Subtract 2429 from 2500, resulting in 71. The quotient is 1.77.

Step 9: Continue these steps until you achieve the desired precision.

The square root of √3.14 is approximately 1.772.

The approximation method is a simple method for finding square roots. Let us learn how to find the square root of 3.14 using this method.

Step 1: Identify the closest perfect squares to 3.14. The closest perfect squares are 1 (1^2) and 4 (2^2). Hence, √3.14 lies between 1 and 2.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula: (3.14 - 1) / (4 - 1) = 2.14 / 3 ≈ 0.713.

Step 3: Adding this decimal value to the smaller perfect square's root gives: 1 + 0.713 ≈ 1.713.

Therefore, the approximate square root of 3.14 is 1.772 (more precise calculations yield this value).

• Square root: The square root is the inverse operation of squaring a number. For example, if 4^2 = 16, then √16 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction or a ratio of two integers. For example, √3.14 is irrational.

• Decimal: A decimal is a way of representing numbers that include a fractional part, separated by a decimal point. Examples include 3.14 and 1.772.

• Approximation: An approximation is a value or number that is close to, but not exact, often used for irrational numbers like √3.14 ≈ 1.772.

• Long division method: This is a technique used to find the square root of non-perfect squares by dividing the number into smaller, manageable parts.