SQUARE ROOTS

About "Square roots"

Square roots :

The square root of a number is the value such that, when a number multiplied by itself,for example

   3 x 3 = 9

It is written with a radical symbol " √ " and the number or expression inside the radical symbol is called the radicand.

Methods to find square-root of perfect square

(i) Prime factorization 

(ii) Long division method

How to find the square-root of a number using prime factorization

• Using this method, first we have to split the given number into prime factors.
• Write those prime factors inside the radical sign instead of the given number.
• Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.
• Multiply the numbers which have come out from the radical sign.
  

Examples

Question 1 : 

Find square-root of 324 by prime factorization

Solution :

Step 1 :

Split 324 into prime factors 

Step 2 :

√324  =  √2 x 2 x 3 x 3 x 3 x 3 

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

Hence, square root of 324 is 18.

Question 2 : 

Find square root of 625 by prime factorization

Solution :

Step 1 :

Split 625 into prime factors 

Step 2 :

√625  =  √5 x 5 x 5 x 5

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

  =  5 x 5

  =  25

Hence, √625 is 25.

Question 3 : 

Find square-root of 4096 by prime factorization

Solution :

Step 1 :

Split 625 into prime factors 

Step 2 :

√4096  =  √2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Step 3 :

Take one common number from the radical

  =  2 x 2 x 2 x 2

  =  64

Hence, square-root of 4096 is 64.

Question 4 : 

Find square-root of 400 by prime factorization

Solution :

Step 1 :

Split 400 into prime factors 

Step 2 :

√400  =  √2 x 2 x 2 x 2 x 5 x 5

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

  =  2 x 2 x 5

  =  20

Hence, square root of √400 is 20.

Question 5 : 

Find square-root of 144 by prime factorization

Solution :

Step 1 :

Split 144 into prime factors 

Step 2 :

√144  =  √2 x 2 x 2 x 2 x 3 x 3

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

  =  2 x 2 x 3

  =  12

Hence, √144  is 12.

Question 6 : 

Find the square-root of 104976 by long division method  

Solution :

Step 1 :

Separate the digits by taking commas from right to left once in two digits.

10,49,76  

When we do so, we get 10 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product ≤ 10

(The product must be greatest and also less than 10)

The above condition will be met by “3”.

Because 3x3 = 9 ≤ 10

Now this situation is explained using long division

In the above picture, 9 is subtracted from 10 and we got the remainder 1.

Step 3 :

Now, we have to bring down 49 and quotient 3 to be multiplied by 2 as given in the picture below.

Step 4 :

Now we have to take a same number at the two places indicated by "?".

Then, we have to find the product as shown in the picture and also the product must meet the condition as indicated.

Step 5 :

The condition said in step 4 will be met by replacing "?" with "2". 

Than we have to do the calculation as given in the picture. 

Step 6 :

Now, we have to bring down 76 and quotient 32 to be multiplied by 2 as given in the picture below.

Step 7 :

In the above picture, we have applied the procedures explained in step 4 and step 5. And we got the remainder zero.  

Step 8 :

From the above picture, finally we got the square root of 104976. That is 324.

Hence, the square root of 104976 = 324

After having gone through the stuff given above, we hope that the students would have understood "Square-roots". 

Apart from the stuff given above, if you want to know more about "Square roots", please click here.

Apart from the stuff given on this web page, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us : 

v4formath@gmail.com

We always appreciate your feedback. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6