Square Root of 112

The square root of 112 is ±10.5830052443.  Basically, finding the square root is just the inverse of squaring a number and hence, squaring 10.5830052443 will result in 112.  The square root of 112 is written as √112 in radical form. In exponential form, it is written as (112)^1/2 
 

We can find the square root of 112 through various methods. They are:

i) Prime factorization method

ii) Long division method

iii) Approximation/Estimation method
 

The prime factorization of 112 is done by dividing 112 by prime numbers and continuing to divide the quotients until they can’t be divided anymore. 

• Find the prime factors of 112

• After factoring 112, 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.

So, Prime factorization of 112 = 2× 2×2×2×7   

But here in case of 112, two pairs of factors can be obtained and a single 7 is remaining. So, it can be expressed as 4√7, the simplest radical form of √112. 
 

This is a method used for obtaining the square root for non-perfect squares. 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 112:

Step 1 : Write the number 112, 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 1. Here, it is 1, Because 1²=1 < 1.

Step 3 : Now divide 1 by 1 such that we get 1 as quotient and then multiply the divisor with the quotient, we get 1

Step 4: Subtract 1 from 1. Bring down 1 and 2 and place it beside the difference 0.

Step 5: Add 1 to the same divisor, 1. We get 2.

Step 6: Now choose a number such that when placed at the end of 2, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 1. Here, that number is 0. 20×0=0<1.

Step 7: Subtract 12-0=12. Add a decimal point after the new quotient 10, again, bring down two zeroes and make 12 as 1200. Simultaneously add the unit’s place digit of 20, i.e., 0 with 20. We got here, 20. 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, 17500 (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 10.58….

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 112. Here, it is 100 and 121.

Step 2: We know that, √100=10 and √121=11. This implies that √112 lies between 10 and 11.

Step 3: Now we need to check √112 is closer to 10 or 11. Let us consider 10.5 and 11. Since (10.5)²=110.25 and (11)²=121. Thus, √112 lies between 10.5 and 11.

Step 4: Again considering precisely, we see that  √112 lies close to (10.5)2=110.25. Find squares of (10.52)²=110.67 and (10.8⁾²= 116.64.

We can iterate the process and check between the squares of 10.55 and 10.75 and so on.

We observe that √112=10.58…
 

• 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, 2⁴ = 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

• Decimal places - It is defined as the digits that appear to the right of the decimal point. Ex- the value of the square root of 115 rounded to 2 decimal places is 10.72