Square Root of 141

The square root is the inverse of the square of the number. 141 is not a perfect square. The square root of 141 is expressed in both radical and exponential form. In the radical form, it is expressed as √141, whereas (141)^(1/2) in the exponential form. √141 ≈ 11.8743, 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 long-division and approximation methods 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 141 is broken down into its prime factors.

Step 1: Finding the prime factors of 141 Breaking it down, we get 3 x 47: 3¹ x 47¹

Step 2: Now we found out the prime factors of 141. Since 141 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √141 using prime factorization is not feasible.

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 141, we need to group it as 41 and 1.

Step 2: Now we need to find n whose square is 1. We can say n is ‘1’ because 1 x 1 is less than or equal to 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 41, which is the new dividend. Add the old divisor with the same number 1 + 1; we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 41. Let us consider n as 1, now 2 x 1 x 1 = 21

Step 6: Subtract 41 from 21; the difference is 20, and the quotient is 11.

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 2000.

Step 8: Now we need to find the new divisor, that is 109, because 219 x 9 = 1971

Step 9: Subtracting 1971 from 2000, we get the result 29.

Step 10: Now the quotient is 11.9

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √141 is approximately 11.87

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 141 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √141. The smallest perfect square less than 141 is 121, and the largest perfect square greater than 141 is 144. √141 falls somewhere between 11 and 12.

Step 2: Now we need to apply the formula that is (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).

Going by the formula (141 - 121) ÷ (144 - 121) = 20/23 ≈ 0.8696 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 11 + 0.8696 ≈ 11.87.

So the square root of 141 is approximately 11.87.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is, √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.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that is more prominently used in the real world. This is why it is also known as the principal square root.
   
• Prime factorization: This is the process of expressing a number as the product of its prime factors.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.