Square Root of 1.42

The square root is the inverse of the square of the number. 1.42 is not a perfect square. The square root of 1.42 is expressed in both radical and exponential form. In the radical form, it is expressed as √1.42, whereas (1.42)^(1/2) in the exponential form. √1.42 ≈ 1.191, 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:

• 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. In the case of 1.42, we start with 1 and then 42.

Step 2: Now we need to find a number whose square is closest to 1. That number is 1, because 1 x 1 is equal to 1. Now the quotient is 1 after subtracting 1 - 1 the remainder is 0.

Step 3: Bring down 42, the new dividend. Add the old divisor with the same number: 1 + 1 = 2, which will be our new divisor.

Step 4: The new divisor will be 2, we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 42. Let us consider n as 1, now 2 x 1 x 1 = 2.

Step 6: Subtract 42 from 2, the difference is 40, and the quotient is 1.1.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 4000.

Step 8: Now we need to find the new divisor. We can use 21 because 211 x 1 = 211.

Step 9: Subtracting 211 from 4000 we get the result 1789. Step 10: Now the quotient is 1.19.

Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So, the square root of √1.42 is approximately 1.191.

The approximation method is another method for finding the 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 1.42 using the approximation method.

Step 1: Now we have to find the closest perfect square to √1.42. The smallest perfect square less than 1.42 is 1, and the largest perfect square more than 1.42 is 4. √1.42 falls somewhere between √1 and √4, i.e., between 1 and 2.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Using the formula, (1.42 - 1) / (4 - 1) = 0.14. Using the formula, we identified the approximate decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 1 + 0.14 = 1.14,

so the square root of 1.42 is approximately 1.14.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 4² = 16, and the inverse of the square is the square root, so √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a fraction of two integers. Examples include √2 and π.

• Radical: In mathematics, a radical is an expression that includes a square root, cube root, etc. Example: √9 is a radical expression.

• Approximation: Approximation is the process of finding a value that is close enough to the correct value, usually within an acceptable error margin. Example: π ≈ 3.14.

• Long division method: This method is used to find the square root of non-perfect squares by repeatedly dividing and averaging over multiple steps until the desired precision is achieved.