Square Root of 32

The square root of 32 is ±5.656854… . Finding the square root is just the inverse of squaring a number and hence, squaring 5.656854… will result in 32.  The square root of 32 is written as √32 in radical form. In exponential form, it is written as (32)^1/2 
 

We can find the square root of 32 through various methods. They are:

•  Prime factorization method

•  Long division method

•  Approximation/Estimation method

The prime factorization of 32 is done by dividing 32 by prime numbers and continuing to divide the quotients until they can’t be divided anymore.

• Find the prime factors of 32

• After factorizing 20, make pairs out of the factors to get the square roo

 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 32 = 2 × 2 × 2 × 2 × 2 

But here in case of 32, two pairs of factor 2 can be obtained and a single 2 is remaining

So, it can be expressed as  √32 =  2 × 2 ×√2 = 4√2

 4√2 is the simplest radical form of √32 
 

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 32:

 Step 1: Write the number 32, 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 32. Here, it is
5, Because 5²=25 < 32.

Step 3: Now divide 32 by 5 (the number we got from Step 2) such that we get 5 as quotient
and then multiply the divisor with the quotient, we get 25

Step 4: Subtract 25 from 32. Add a decimal point after the quotient 5, and bring down two zeroes and place it beside the difference 7 to make it 700.

Step 5: Add 5 to same divisor, 5. We get 10.

Step 6: Now choose a number such that when placed at the end of 10, a 3-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 700. Here, that number is 6. 

106×6=636<700.

Step 7: Subtract 700-636=64. Again, bring down two zeroes and make 64 as 6400. Simultaneously add the unit’s place digit of 106, i.e., 6 with 106. We get here, 112. 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, 77500 (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 5.65….

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 32. Here, it is 25 and 36.

Step 2: We know that, √25=5 and √36=6. This implies that √32 lies between 5 and 6.

Step 3: Now we need to check √32 is closer to 5 or 6. Let us consider 5.5 and 6. Since (5.5)²=30.25 and (6)²=36.

Thus, √32 lies between 5.5 and 6.

Step 4: Again considering precisely, we see that  √32 lies close to (5.5)²=30.25. Find squares of (5.6)²=31.36 and (5.8)²= 33.64.
 

We can iterate the process and check between the squares of 5.62 and 5.7 and so on.

We observe that √32=5.65…

 

• 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

• 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 32 rounded to 2 decimal places is 5.65