Square Root of 12

The square root of 12 is ±3.4641016151377544.  Basically, finding the square root is just the inverse of squaring a number and hence, squaring 3.4641016151377544 will result in 12.  The square root of 12 is written as √12 in radical form. In exponential form, it is written as (12)^1/2 
 

We can find the square root of 12 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method
   

The prime factorization of 12 is done by dividing 12 by prime numbers and continuing to divide the quotients until they can’t be divided anymore. 

• Find the prime factors of 12

• After factorizing 12, make pairs out of the factors to get the square root.

•  If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.
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So, Prime factorization of 12 = 2 × 2 × 3  

But here in case of 12, a pair of factor 2 can be obtained and a single 3 is remaining

So, it can be expressed as  √12 =  2 × √3 = 2√3

 2√3 is the simplest radical form of √12 

This is a method used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

Follow the steps to calculate the square root of 12:

Step 1 : Write the number 12, and draw a horizontal bar above the pair of digits from right to left.

Step 2 : Now, find the greatest number whose square is less than or equal to 12. Here, it is 3, Because 3²=9 < 12.

Step 3 : Now divide 12 by 3, such that we get 3 as a quotient and then multiply the divisor with the quotient, we get 9.

Step 4: Add a decimal point after the quotient 3, and bring down two zeroes and place it beside the difference 3 to make it 300.

Step 5: Add 3 to the same divisor, 3. We get 6.

Step 6: Now choose a number such that when placed at the end of 6, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 300. Here, that number is 4. 
64×4=256<300.

Step 7: Subtract 300-256=44. Again, bring down two zeroes and make 44 as 4400. Simultaneously add the unit’s place digit of 64, i.e., 4 with 64. We get here, 68. Apply Step 5 again and again until you reach 0. 

We will show two places of precision here, and so, we are left with the remainder, 704 (refer to the picture), after some iterations and keeping the division till here, at this point 

             
Step 8 : The quotient obtained is the square root. In this case, it is 3.464….

Approximation or estimation of square root is not the exact square root, but it is an estimate. Here, through this method, an approximate value of square root is found by guessing.

Follow the steps below:

Step 1: Find the nearest perfect square number to 12. Here, it is 9 and 16.

Step 2: We know that, √9=3 and √16=4. This implies that √12 lies between 3 and 4.

Step 3: Now we need to check √12 is closer to 3 or 4. Let us consider 3 and 3.5. Since (3)²=9 and (3.5)²=12.25. Thus, √12 lies between 3 and 3.5.

Step 4: Again considering precisely, we see that  √12 lies close to (3.5)²=12.25. Find squares of (3.2)²=12.24 and (3.49)²= 12.18.

We can iterate the process and check between the squares of 3.4 and 3.48 and so on.

We observe that √12=3.464…





 

• Exponential form:  An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 2 4 = 16, where 2 is the base, 4 is the exponent 

• Prime Factorization:  Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3

• Prime Numbers: Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

• Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

• Perfect and non-perfect square numbers: Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24