Square Root of 1.96

The square root is the inverse operation of squaring a number. 1.96 is a perfect square. The square root of 1.96 can be expressed in both radical and exponential form. In radical form, it is expressed as √1.96, whereas in exponential form, it is (1.96)^(1/2). √1.96 = 1.4, which is a rational number because it can 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. For non-perfect square numbers, methods like the long-division method and approximation method are used. However, since 1.96 is a perfect square, let's proceed with the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Let's see how 1.96 is broken down into its prime factors.

Step 1: Express 1.96 as a fraction: 1.96 = 196/100.

Step 2: Find the prime factors of 196 and 100. 196 = 2 x 2 x 7 x 7 100 = 2 x 2 x 5 x 5

Step 3: Taking the square root of both the numerator and the denominator: √(196/100) = √196 / √100 = (2 x 7) / (2 x 5) = 14/10 = 1.4

The long division method is particularly useful for finding square roots of non-perfect squares, but it can also be applied to perfect squares like 1.96 for precision.

Step 1: Set up 1.96 for long division and group it as 1.96.

Step 2: Find a number whose square is close to 1. The closest perfect square is 1, so the initial quotient is 1.

Step 3: Bring down the next pair (96), making the new dividend 96. Double the initial quotient to use as a new divisor, which is now 20.

Step 4: Find the largest digit (n) such that 20n x n is less than or equal to 96. The number is 4, since 204 x 4 = 816.

Step 5: Subtract 816 from 960, which leaves you with 144.

Step 6: Add a decimal point and bring down two zeros, making the new dividend 14400.

Step 7: Double the current quotient to get a new divisor, which becomes 28. Find a digit n such that 28n x n is less than or equal to 14400. The correct digit is 5, as 285 x 5 = 1425.

Step 8: The quotient is now 1.4.

The approximation method is another way to find square roots, particularly useful for non-perfect squares, but we can apply it for quick checks.

Step 1: Identify the closest perfect squares around 1.96, which are 1 (1^2) and 4 (2^2). √1.96 falls between 1 and 2.

Step 2: Use linear interpolation to estimate more accurately. Since 1.96 is closer to 2 than 1, we arrive at 1.4 by checking values or using a calculator.

Students often make mistakes while finding square roots. Common errors include neglecting the negative square root or misapplying the division method. Let's explore these mistakes in detail.

• Square root: A square root is the inverse of squaring a number. Example: 1.4^2 = 1.96, and the inverse is √1.96 = 1.4.
   
• Rational number: A rational number is a number that can be expressed in the form of p/q, where q is not zero and p and q are integers.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 4 is a perfect square because it is 2^2.
   
• Decimal: A decimal is a number that has a whole number part and a fractional part separated by a decimal point, such as 1.96.
   
• Linear interpolation: A method used to approximate values between two known values, often used in estimation.