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Last updated on December 2nd, 2024

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Square Root of 32

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Foundation
Intermediate
Advance Topics

The square root of 32 is a value โ€œyโ€ such that when โ€œyโ€ is multiplied by itself โ†’ y โคซ y, the result is 32. The number 32 has a unique non-negative square root, called the principal square root.

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What Is the Square Root of 32?

The square root of 32 is ยฑ5.656854โ€ฆ . Finding the square root is just the inverse of squaring a number and hence, squaring 5.656854โ€ฆ will result in 32.  The square root of 32 is written as โˆš32 in radical form. In exponential form, it is written as (32)1/2 
 

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Finding the Square Root of 32

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

 

  •  Prime factorization method

 

  •  Long division method

 

  •  Approximation/Estimation method
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Square Root of 32 By Prime Factorization Method

The prime factorization of 32 is done by dividing 32 by prime numbers and continuing to divide the quotients until they canโ€™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 โ€œradicalโ€ sign along with the obtained pairs.

So, Prime factorization of 32 = 2 ร— 2 ร— 2 ร— 2 ร— 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  โˆš32 =  2 ร— 2 ร—โˆš2 = 4โˆš2


 4โˆš2 is the simplest radical form of โˆš32 
 

 

 

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Square Root of 32 By Long Division Method

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. 


106ร—6=636<700.


Step 7: Subtract 700-636=64. Again, bring down two zeroes and make 64 as 6400. Simultaneously add the unitโ€™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 5.65โ€ฆ.

 

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Square Root of 32 By Approximation

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, โˆš25=5 and โˆš36=6. This implies that โˆš32 lies between 5 and 6.

 

 

Step 3: Now we need to check โˆš32 is closer to 5 or 6. Let us consider 5.5 and 6. Since (5.5)2=30.25 and (6)2=36.

 

Thus, โˆš32 lies between 5.5 and 6.

 

 

Step 4: Again considering precisely, we see that  โˆš32 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 โˆš32=5.65โ€ฆ

 

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Important Glossaries for Square Root of 32

  • 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 ร— 2 ร— 2 ร— 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 ร— 2 ร— 2 ร— 2 ร— 3

 

  • Prime Numbers: Numbers which are greater than 1, having only 2 factors as โ†’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
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