Square and square root

Learning Outcomes:

After reading this post readers will be able to,

• Define what is a perfect square.

• Identify a number is a perfect square or not.

• Recognize the properties of perfect square.

• Find the square root of the nature numbers by factorization method and division method.

• difference between square and square root

Introduction:

We know that the area of a square can be calculated by multiplying length of square by itself as 

Area of square = length × length

Suppose a square has a length "x" then

Area of square = (x) × (x) = (x)²

It means x² is a area of square whose length is x or simply we can say that x² is the square of x . i.e The square of x = x²

Thus the square of a number can be defined as below:

The product of a number with itself is called square of it.

Perfect Square:

Any natural number is called a perfect Square, if it is the square of any natural number. To make it clear lets find the square of some natural numbers.

1² = 1 × 1 = 1

2² = 2 × 2 = 4

3² = 3 × 3 = 9

4² = 4 × 4 = 16

5² = 5 × 5 = 25

6² = 6 × 6 = 36

7² = 7 × 7 = 49

8² = 8 × 8 = 64

9² = 9 × 9 = 81

10² = 10 × 10 = 100 and so on.

Hence 1 is the square of 1.

4 is the square of 2.

9 is the square of 3.

16 is the square of 4 and so on.

To check a number is a perfect Square or not:

To check a number is perfect Square or not, write the number as a product of its prime factors of all factors can be grouped in pairs then the given number is a perfect square.

Example: check the following numbers are perfect square or not.

1) 3969  

2) 6084

3) 3873

Solution:

1) prime factors of 3969 = 3×3 ×3×3 × 7×7

We note that each factor form a pair. So 3969 is perfect square.

2) prime factors of 6084 = 2×2 × 3×3 × 13×13

Each factor of 6084 from a pair. So it is a perfect square.

3) prime factors of 3872 = 2×2 × 2 × 2×2 × 11×11

We note that 2 is a factor which can't be paired with any equal factor. So 3872 is not a perfect square.

Properties of perfect squares:

1) The square of an even number is even:

As we know the natural numbers are divided into two groups, even numbers and odd numbers.

See the following squares,

2² = 2×2 = 4

4² = 4×4 = 16

6² = 6×6 = 36

8² = 8×8 = 64

10² = 10×10 = 100

12² = 12×12 = 144

We note that the squares of all even numbers are even numbers.

2) Square of an odd number is odd:

See following examples,

1² = 1×1 = 1

3² = 3×3 = 9

5² = 5×5 = 25

7² = 7×7 = 49

9² = 9×9 = 81

So the squares of odd numbers are also odd numbers.

3) The square of a proper fraction is less than itself:

To Square a proper fraction we multiply the numerator and denominator by itself.

For example (2/5)

We take square of it as (2/5)²

= (2/5) × (2/5) = (2×2)/(5×5) = 4/25

Now we compare the fraction 2/5 with its square 4/25

2/5 , 4/25

Take L.C.M of 5 and 25 which is 25 and multiply it to both the fractions

(2/5)×25 , (4/25)×25

50 , 20

So 50 > 20

so it is observe that the square of a proper fraction is less than itself.

so 2/5 > 4/25

Similarly (1/3)² = 1/3 x 1/3 = 1/9

so 1/3 > 1/9

4) Square of a decimal less than 1 is smaller than the decimal:

To find the square of decimal, we use following method.

(0.4)² = (0.4) × (0.4) = 0.16

Is 0.16 is smaller than 0.4 or greater ? Certainly 0.16 is smaller than 0.4 

so 0.16 < 0.4

it means square of a decimal is always smaller than the given decimal.

Square Root:

Process of finding square root is an opposite operation of squaring a number.

 To understand this we find some perfect squares.

2² = 4 ( 2 squared is 4)

5² = 25 ( 5 squared is 25)

7² = 49 ( 7 squared is 49)

The above equations can be read as 

2 is the square root of 4.

5 is the square root of 25 and

7 is the square root of 49.

Similarly we can find the square root of any square number. For this we use a symbol ( √ ) which represent a square root. For example,

√(y²) = y where "√" is called radical sign. Here y² is called radicand.

Square root by prime Factorization:

As we know that:

The square root of 4 is , √4 = √(2²) = 2

The square root of 9 is , √9 = √(3²) = 3

The square root of 25 is , √25 = √(5²) = 5

And so on. But in large perfect squares it is difficult for us to guess their square roots. So we used an other method which is called prime factorization method. To understand this method see following example.

Suppose the number is 36

36 = 2×2×3×3

Take square root on both side.

√(36) = √(2×2×3×3)

Write them in pair of prime factors of a perfect square.

√36 = √(2×2) × √(3×3)

√36 = √2² × √3²

So √36 = 2 × 3 = 6 Ans.

Finding square roots of fractions:

As we know that there are three types of common fractions.

1) proper fraction

2) Improper fraction

3) Compound fraction or mixed fraction

Example: square root of a common fraction (144/256)

Solution:

Ist we take square root of (144/256).

Example: An improper fraction (144/81), find the square root.

Solution:

Square roots of decimals:

In decimals first of all we change them into common fractions and then find square root. After finding square root we write the answer in decimal form again. To understand this we see an example.

Example: Find the square root of 0.64.

Solution:

First we change decimal into fraction as 0.64 = 64/100

Now we take square root of it as √(64/100)

Factorization of 64 = 2×2×2×2×2×2

Factorization of 100 = 2×2×5×5

Square root by division method: 

In previous example we have learnt the process to find square root of natural numbers by Prime factorization method but now we learn the square root of natural numbers by division method.

Example: By division method find the square root of 324.

Solution:

Example: By division method find the square root of 585225. 

Solution:

Example: By division method find the square root of (4096/15129).

Solution:

Have a nice day.