SQUARE ROOTS OF PERFECT SQUARE

About "Finding square roots of perfect squares"

Finding square roots of perfect squares :

Finding square roots is the opposite process squaring. 

1²  =  1, the square root of 1 is 1.

2²  =  4, the square root of 4 is 2.

3²  =  9, the square root of 9 is 3.

4²  =  16, the square root of 16 is 4.

what is perfect square ?

From the above example, we come to know that 16, 9, 4 and 1 are known as perfect squares. Because we can represent all as the multiples two same numbers.

That is 16 = 4 x 4, 9 = 3 x 3, 4 = 2 x 2, 1 = 1 x 1.

Like wise 18 is not a perfect square, because we cannot represent this 18 as the multiple of two same terms.  

Usually we follow two methods to find square roots of perfect squares.

  • Prime factorization method
  • Long division method

Let us see some example problems to understand the method of finding square root using the above methods.

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.

Question 1 :

Find the prime factors of 324

Solution :

Step 1 :

Since the given number ends with 4, first we have to split the given number by the smallest even prime number 2.

Step 2 :

2 goes into 3 one time.We have 1 left. If we take this 1 along with the next digit 2, we get 12. If we divide this by 2, we get 6. 

We don’t have any number remaining in 12. So we can take the next digit 4. Again, if we divide 4 by 2, we get 2.

Step 3 :

324 = √(2 x 2 x 3 x 3 x 3 x 3)

  =  2 x 3 x 3

  =  18

Hence the square square root of 324 is 18.

We can do the same problem using long division method

Step 1 :

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

3, 24

When we do so, we get 3 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product  3

(The product must be greatest and also less than 3)

The above condition will be met by “1”.

Because 1 x 1 = 1  3, but 2 x 2 = 4 > 3

Now this situation is explained using long division

In the above picture, 1 is subtracted from 3 and we got the remainder 2.

Step 3 :

Now, we have to bring down 24 and quotient 1 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. 

Hence the square square root of 324 is 18.

Question 2 :

Find the prime factors of 625

Solution :

Step 1 :

Since the given number ends with 5, first we have to split it by the prime number 5.

Step 2 :

5 goes into 6 one time.We have 1 left. If we take this 1 along with the next digit 2, we get 12. Again we have to divide it by 5. If we divide this by 5, we get 2.

Now we have 2 left. Now we have to take this 2 along with the next digit 5, we get 25.  If we divide 25 by 5, we get 5.

Step 3 :

By repeating this process until we get prime factors. 

Hence, 625 = √(5 x 5 x 5 x 5)  =  5 x 5  = 25

We can do the same problem using long division method

Step 1 :

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

3, 24

When we do so, we get 3 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product  6

(The product must be greatest and also less than 6)

The above condition will be met by “1”.

Because 2 x 2 = 4  6, but 3 x 3 = 9 > 6

Now this situation is explained using long division

In the above picture, 4 is subtracted from 6 and we got the remainder 2.

Step 3 :

Now, we have to bring down 25 and quotient 1 to be multiplied by 2.

Now we have put a number next to 4, and the same number should be in the quotient next to 2. So that their product must be less than or equal to 225.

By multiplying 5 and 45, we will get the answer 225.

Hence 25 is the square root of 625.

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

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

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

HTML Comment Box is loading comments...

ALGEBRA

Variables and constants

Writing and evaluating expressions

Solving linear equations using elimination method

Solving linear equations using substitution method

Solving linear equations using cross multiplication method

Solving one step equations

Solving quadratic equations by factoring

Solving quadratic equations by quadratic formula

Solving quadratic equations by completing square

Nature of the roots of a quadratic equations

Sum and product of the roots of a quadratic equations 

Algebraic identities

Solving absolute value equations 

Solving Absolute value inequalities

Graphing absolute value equations  

Combining like terms

Square root of polynomials 

HCF and LCM 

Remainder theorem

Synthetic division

Logarithmic problems

Simplifying radical expression

Comparing surds

Simplifying logarithmic expressions

Negative exponents rules

Scientific notations

Exponents and power

COMPETITIVE EXAMS

Quantitative aptitude

Multiplication tricks

APTITUDE TESTS ONLINE

Aptitude test online

ACT MATH ONLINE TEST

Test - I

Test - II

TRANSFORMATIONS OF FUNCTIONS

Horizontal translation

Vertical translation

Reflection through x -axis

Reflection through y -axis

Horizontal expansion and compression

Vertical  expansion and compression

Rotation transformation

Geometry transformation

Translation transformation

Dilation transformation matrix

Transformations using matrices

ORDER OF OPERATIONS

BODMAS Rule

PEMDAS Rule

WORKSHEETS

Converting customary units worksheet

Converting metric units worksheet

Decimal representation worksheets

Double facts worksheets

Missing addend worksheets

Mensuration worksheets

Geometry worksheets

Comparing  rates worksheet

Customary units worksheet

Metric units worksheet

Complementary and supplementary worksheet

Complementary and supplementary word problems worksheet

Area and perimeter worksheets

Sum of the angles in a triangle is 180 degree worksheet

Types of angles worksheet

Properties of parallelogram worksheet

Proving triangle congruence worksheet

Special line segments in triangles worksheet

Proving trigonometric identities worksheet

Properties of triangle worksheet

Estimating percent worksheets

Quadratic equations word problems worksheet

Integers and absolute value worksheets

Decimal place value worksheets

Distributive property of multiplication worksheet - I

Distributive property of multiplication worksheet - II

Writing and evaluating expressions worksheet

Nature of the roots of a quadratic equation worksheets

Determine if the relationship is proportional worksheet

TRIGONOMETRY

SOHCAHTOA

Trigonometric ratio table

Problems on trigonometric ratios

Trigonometric ratios of some specific angles

ASTC formula

All silver tea cups

All students take calculus 

All sin tan cos rule

Trigonometric ratios of some negative angles

Trigonometric ratios of 90 degree minus theta

Trigonometric ratios of 90 degree plus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 180 degree minus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 270 degree minus theta

Trigonometric ratios of 270 degree plus theta

Trigonometric ratios of angles greater than or equal to 360 degree

Trigonometric ratios of complementary angles

Trigonometric ratios of supplementary angles 

Trigonometric identities 

Problems on trigonometric identities 

Trigonometry heights and distances

Domain and range of trigonometric functions 

Domain and range of inverse  trigonometric functions

Solving word problems in trigonometry

Pythagorean theorem

MENSURATION

Mensuration formulas

Area and perimeter

Volume

GEOMETRY

Types of angles 

Types of triangles

Properties of triangle

Sum of the angle in a triangle is 180 degree

Properties of parallelogram

Construction of triangles - I 

Construction of triangles - II

Construction of triangles - III

Construction of angles - I 

Construction of angles - II

Construction angle bisector

Construction of perpendicular

Construction of perpendicular bisector

Geometry dictionary

Geometry questions 

Angle bisector theorem

Basic proportionality theorem

ANALYTICAL GEOMETRY

Analytical geometry formulas

Distance between two points

Different forms equations of straight lines

Point of intersection

Slope of the line 

Perpendicular distance

Midpoint

Area of triangle

Area of quadrilateral

Parabola

CALCULATORS

Matrix Calculators

Analytical geometry calculators

Statistics calculators

Mensuration calculators

Algebra calculators

Chemistry periodic calculator

MATH FOR KIDS

Missing addend 

Double facts 

Doubles word problems

LIFE MATHEMATICS

Direct proportion and inverse proportion

Constant of proportionality 

Unitary method direct variation

Unitary method inverse variation

Unitary method time and work

SYMMETRY

Order of rotational symmetry

Order of rotational symmetry of a circle

Order of rotational symmetry of a square

Lines of symmetry

CONVERSIONS

Converting metric units

Converting customary units

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