Square Root of 142

The value that results from multiplying a number by itself is called its square. For instance, the square of 5 is 5x5 = 25. Now we know 25 is the square of 5. Next, we can learn more about the square root of 142. It is the number that yields 142 when multiplied by itself. The square root of 142 is expressed as √142  in radical sign, or 142^1/2  in exponential form. Now, we can delve deeper into this crucial concept.  

Square Root of 142:

Calculating a number's square root is quite simple. The square root of 142 is a unique number that, when multiplied by itself, gives 142. It is not a perfect square. Take a look at this:
142= 11.9164

Here, you just imagine that 11.9164 times 11.9164, we will come close to 142. So, finding the square root is like figuring out the number that makes 142, when we multiply it by itself. Keep in mind that our answer can be a decimal or a whole number.

Square Root of 142 in exponential form:
When we talk about exponential form, the square root of 142 is represented as a power of 12 . Instead of using the radical sign, we use 12  to show the square root of 142. The square root calculation can be written as a number, called the base, with a fractional exponent. Let us write it down:

√b = b^1/2
142^1/2 = 11.9164

This means the square root of 142 is equal to the power of  1/2 . 

Square Root of 142 in radical form:
The square root of 142 is represented as √142  in radical form. This means, if we multiply the square root of 142 by itself the answer will be 142.

To calculate the square root of 142, we have to identify the number, which is multiplied by itself and gives the original number. See, the square root of 142 is considered as q, it will be like:

142 = q x q = q²

One thing you should remember is that every positive real number has two square roots. That can be a positive and a negative square root. The unique positive value is known as the principal square root or principal root.

There are different methods we use to find the square of 142. The most commonly used are as follows:

• Prime Factorization
• Long division
• Approximation 

We’ll now look into each of these in detail.
 

To find perfect squares, we use the prime factorization method but as we know, 142 is not a perfect square.

However, we can try the prime factorization to find the square root of 142. In this method, we will break 142 into its prime numbers.  For this, we have to follow certain steps. 

Step 1: Break 142 into its prime numbers. 

Since 142 is an even number, as we know, it may be divided by two. Therefore, we begin with 2, the smallest prime number.

142 / 2 = 71

Step 2: Find the prime factorization of 71.

71 is a prime number, and its prime factorization is 71¹. Hence, the prime factorization of 142 is 2¹x71^1.

Step 3: Check for pairs.

In the prime factorization method, we need pairs of the same number to find the square root. But there are no pairs in 2x 71. Since 142 doesn’t have any pairs, it is difficult to find the square root by using prime factorization.   

If the number has pairs of prime factors, the prime factorization method helps to find out the square root. 
 

The long division method provides an appropriate value for the square root of 142 since it is a decimal. It is a step-by-step process. Let us take a look at the various steps of the long division method.

Step 1: Break 142 into two pairs of two digits from right to left. Additionally, add a set of 00 at the end because we want one decimal.

1   42   00 

 Step 2: Figure out a number for ’n’. Take the number 1, because it is the largest perfect square less than or equal to 1. Now, we can apply it on n² ≤ 1. 
 1² ≤ 1 which simplifies to  1≤ 1. This shows the quotient is 1 and the remainder will be 0. 

Step 3: 2n is the next divisor. Here, ‘n’ is the previous divisor. So, 2 x 1 = 2.

Step 4: Now, move on to the next set of numbers. That is 42. Thus, the new dividend is 42.

Step 5: Next,  figure out a number for ‘Y’, such that 2Y  Y  42. Let us take 2 as Y. 

22 x  2 = 44. It is greater than 42. 

So, we can move to 1. 

21 x  1 = 21. It is less than 42. 

Step 6: Subtract 21 from 42. The result is 21 and the quotient is 11. 

Step 7:  In this stage, we took the two zeros, which we added in the first step. Now, the remainder is 2100. Next, double the quotient, i.e., 11, which gives 22. So, 22 is the next divisor. 

Step 8: Find the digit for ‘Y’. 22Y  x Y  < 2100. Let us take 9.

229 x  9 = 2016

Step 9: We will repeat the steps until we get four decimal places for the square root of 142.

 The square root of 142 = 11.9163.
 

In the approximation method, we find the square root of 142 by identifying the closest perfect squares. This is a simple yet effective method to figure out the square root of 142. This method follows certain steps:

Step 1: Find out the closest perfect squares to 142 . 
121 is the smallest perfect square and 144 is a large perfect square compared to 142. √142  lies between 11 and 12.

 
Step 2: Now, we can apply a formula, that is, subtract the smallest perfect square from the given number and the next largest perfect square. Then, divide the subtracted values. To understand clearly, take a look at this:

(142 - 121) / (144 - 121) = 0.913

Step 3:  Next, add the square of 121 to the decimal number. The initial number of the square root of 142 will be 11. 
11 + 0.913 = 11. 913 

Hence, √142 =  approximately 11.913 

• Square root: It is a mathematical value. When you multiply the value by itself, it equals the original number. For example, the square root of 142 is 11.916.

• Perfect square: A perfect square is a number that can be the square of an integer. For instance, 11x  11= 121, is a perfect square. 

• Radical sign:  This is a symbol that is used to represent the square root of any number. √ is the symbolic representation of the square root. √142  Is 11.916. 

• Principal root: Every positive real number has two square roots. That can be a positive and a negative square root. The unique positive value is known as the principal square root or principal root. For example, 11.916 is the principal root of 142.