Hello dear, your most welcome here. Here you will get your solution easily. As you know that mathematics is very important part of our life. In mathematics problems many times we have need to find the Square and Square Roots of numbers. This page is fully dedicated to explain about Square and Square Roots (Explained with Example).
We will discuss about these topics steps by steps and also learn with examples. So read full article for better learning. So lets learn about Square and Square Roots (Explained with Example)…
Square and Square Roots (Explained with Example)
What is square :
If we multiply a number with the number itself then the obtained number is called the square of the number. Let a number y, then the square of y is y × y and denoted as y2 .
Examples-
- 32 = 3 × 3 = 9
- 52 = 5 × 5 = 25
- 122 = 12 × 12 = 144
Perfect square or square number :
Perfect square or Square numbers are those numbers, which can be represented as the product of pairs of equal factors. OR
If any number is the square of a natural number then that number is called the perfect square or a square number.
Examples – 1 = 12, 4 = 22 , 9 = 32 , 16 = 42, 25 = 52, 36 = 62 and so on.
In the above cases, 1, 4, 9,16, 25, 36,….. are square numbers or perfect squares.
Note- A perfect square can always be expressed as the product of pairs of equal factors.
Example 1- Check whether 144 is a perfect square or not. If yes, then find the number whose square is 196.
Solution- First, on resolving 144 into prime factors, we have 144 = 2 × 2 × 2 × 2 × 3 × 3
You can see that 144 can be expressed as the product of pairs of equal factors as we discussed above. Therefore 144 is a perfect square number.
Hence 144 = 22 × 22 × 32
= (2 × 2 × 3)2
= (12)2
Hence 12 is the number whose square is 144. Here 12 is the square root of 144. We will learn about square roots further.
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Example 2- Is, 1764 is a perfect square, find that number whose square is 1764.
Solution- On resolving 1764 into prime factors, we have 1764 = 2 × 2 × 3 × 3 × 7 × 7
You can see that 1764 can be expressed as the product of pairs of equal factors.
Therefore 1764 is a perfect square number.
Hence, 1764 = 22 × 32 × 72
= (2 × 3 × 7)2 = (42)2
Therefore, 42, is the number whose square is 1764.
Example 3- Check whether 2548 is a perfect square.
Solution- On resolving 2548 into prime factors, we have 2548 = 2 × 2 × 7 × 7 × 13 = (22 × 72 × 13)
You can see that 2 and 7 have pairs but 13 has no pair, Hence 2548 can not be expressed as a product of pairs of equal factors. Hence, 6292 is not a perfect square.
Square and Square Roots (Explained with Example)…..
Properties of perfect square numbers :
Here are some properties related to perfect square numbers. These properties help you to find that the given numbers can be a perfect square or not. Read these properties carefully.
Property 1- Any number ending with digitals, 2, 3, 7 or 8 is never a perfect square.
Example- The numbers 72, 83, 167, 238 etc. We see that these numbers are end with digits 2, 3, 7, 8, respectively, hence this type of numbers are not perfect square numbers.
Property 2- If any number ends with an odd number of zeros, then it is never a perfect square.
Example- The numbers 180, 3000, 800000, etc. end with odd numbers of zero (Here number of zeroes are one, there, five ). Therefore this type of numbers are not perfect square.
Property 3- If a number leaves a remainder 2 after dividing by 3, then it is not a perfect square.
Example – 38, 77, 101, 308, 596 etc. If we divide these numbers from 3, we will get 2 as reminder, hence this type of number can not be a perfect square.
Property 4- If a number leaves a remainder 2 or 3 after dividing by 4, then it is not a perfect square.
Example- 578, 654, 798, 1002, etc. If we divide these numbers from 4, we will get either 2 or 3 as reminder, hence this type of number can not be a perfect square.
Property 5- The square of each even number is even.
Example-
22 = 4, 42 = 16 , 62 = 36 , 82 = 64 , 102 = 100 etc.
Hence if square an even number we will always get an even number.
Property 6- The square of each odd number is odd.
Example- 32 = 9 , 52 = 25 , 72 = 49 , 92 = 81 , etc.
Hence if square an odd number we will always get an odd number.
Property 7- The square of each proper fraction is always smaller than the fraction.
Example – ( 3 / 5)2 = 9 / 25 , here (9 / 25) < (3 / 5).
Property 8-
For each natural number n, we have –
(n+1)2 – n2 = (n + 1 + n)(n + 1 – n) = (2n + 1) or {(n + 1) + n}.
Therefore we can write {(n + 1)2 – n2} = {(n + 1) + n}.
That means the difference of the square of two consecutive numbers will be the sum of the numbers. Lets see some examples-
Examples- (i) {(46)2 – (45)2} = (46 + 45) = 91 (ii) {(79)2 + (78)2} = (79 +78) = 157
Property 9- The sum of first n odd natural numbers will be equal to n2 .
Examples- (i) Sum of first 7 odd natural numbers = (1 + 3 + 5 + 7 + 9 + 11 + 13) .
As you are seeing, There are 7 numbers here, Hence n = 7
then the sum of (1 + 3 + 5 + 7 + 9 + 11 + 13 ) = 72 = 49.
(ii) Sum of first 9 odd natural numbers = (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17)
As you are seeing, There are 9 numbers here, Hence n = 9
then the sum of (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17) = 92 = 81.
Property 10 (Pythagorean Triplets)- If there are three natural numbers m, n, p, such that (m2 + n2) = p2, then (m, n, p) are called pythagorean triplet.
Means if we have three natural numbers such that square of greatest number is equal to the sum of the square of other two numbers then numbers are called pythagorean triplet.
Important Result :
If m is a natural number such that m > 1, then for every natural number we have (2m, m2 – 1, m2 + 1) as a pythagorean triplet.
Example – (i) On putting m = 3 in (2m, m2 -1, m2 +1), we get (6, 8, 10) as a pythagorean
triplet.
(ii) On putting m = 6 in (2m, m2-1, m2 +1), we get (12, 35, 37) as a pythagorean
triplet.
Property 11 – There are 2n non-perfect square numbers between two consecutive square numbers n2 and (n+1)2.
That is, if you want to find total number of non perfect square number between two consecutive perfect square number then multiply square root of smaller number by two.
Let’s understand it by example-
Example – (i) Let squares of two consecutive numbers 2 and 3 are 4 and 9 respectively. Here n = 2.
Then the number of non – perfect square numbers between 4 and 9 are 2 × 2 = 4.
(ii) Let squares of two consecutive numbers 7 and 8 are 49 and 64 respectively. Here n = 7.
Then the number of non – perfect square numbers between 49 and 64 are 2 × 7 = 14.
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Square and Square Roots (Explained with Example)…
Shortcut method for the squaring a number :
We had discussed about the square of any number. Let’s discuss the few Shortcut method for the squaring a number .
First we understand Column Method For Squaring the Two Digit Number.
Column Method For Squaring the Two Digit Number :
We’ll understand Column Method using few steps.
Let us consider a number which has the tens digit ‘a’ and the unit digit ‘b’. Now we shall square this number.
Step 1.
Make three columns, I, II and III, headed by a², (2 × a × b) and b² respectively. Write the values of a2, (2 × a × b) and b2 in columns I, II and III respectively.
Step 2.
In Column III, underline the unit digit of b² and carry over the tens digit of it to Column II and add it to the value of (2 × a × b).
Step 3.
In Column II, underline the unit digit of the number which is obtained in Step 2 and carry over the tens digit of it to Column I and add it with the value of a².
Step 4.
Underline this number which was obtained in Step 3 in Column I. The digits which are underlined give the required square number.
To understand these steps we will take few examples.
EXAMPLE 1–
Find the square of 47.
Solution : Given number = 47
Therefore, a = 4 and b = 7.
We will apply the steps given above. First we make a table having columns I, II, & III as shown below . We have a = 4 and b = 7. Write a2 = 42 = 16 in first column, 2ab = 2 × 4 × 7 = 56 in second column and b2 = 72 = 49 in third column. Now we underline unit digit 9 in third column and take tens digit 4 as carry and add with 56 in second column. Here we get 56 + 4 = 60.
Similarly we underline unit digit 0 in second column and take tens digit 6 as carry and add with 16 in first column. Here we get 16 + 6 = 22. Now have to underline both digits of 22 as shown below.
Now we take the numbers having underlined, which are 22, 0 and 9.
Therefore Square of 47 = (47)2 = 2209
I hope you understand these steps . For more clarification , let’s take another example.
EXAMPLE 1–
Find the square of 86.
Solution : Given number = 86
∴ a = 8 and b = 6.
We will apply the steps given above. First we make a table having columns I, II, & III as shown below . We have a = 8 and b = 6. Write a2 = 82 = 64 in first column, 2 × a × b = 2 × 8 × 6 = 96 in second column and b2 = 62 = 36 in third column. Now we underline unit digit 6 in third column and take tens digit 3 as carry and add with 96 in second column. Here we get 96 + 3 = 99.
Similarly we underline unit digit 9 in second column and take tens digit 9 as carry and add with 64 in first column. Here we get 64 + 9 = 73. Now have to underline both digits of 73 as shown below.
Now we take the numbers having underline, which are 73, 9 and 6. Therefore Square of 86 = (86)2 = 7396
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Diagonal Method For the Squaring a Number :
We had discussed about the shortcut method to find square of any number. Let’s discuss the Diagonal Method For the Squaring a Number .
To understand Diagonal Method we are taking an example and will solve it step by step.
Let we have to find the square of 39 by diagonal method.
Solution :
Step 1.
The given number contains two digits. So, draw a square and divide it into 4 subsquares as shown below. Write the digits 3 and 9 vertically and horizontally, as shown below.
Step 2.
Multiply one by one each digit on the left side of the square by each digit on the top. Write the obtained product in the relevant subsquare. If the product is a one-digit number, write it below the diagonal and put 0 above the diagonal.
If we multiply 3 and 3 we will get 9 as product. Since it is a one digit number, therefore we’ll write 9 below the diagonal and put 0 above the diagonal.
In case the product is a two-digit number, write the tens digit above the diagonal and the units digit below the diagonal.
If we multiply 3 and 9 we will get 27 as product. Since it is a two digit number, therefore we’ll write 7 below the diagonal and put 2 above the diagonal.
Do this further for all subsquare.
Step 3.
Start from the lowest diagonal, and add the digits diagonally. If the sum is a two-digit number, underline the units digit and carry over the tens digit to the next diagonal.
In lowest diagonal we have only 1, after that in next diagonal we have 7 + 8 + 7 = 22 , so we underline unit digit 2 and carry the tens digit 2 in the next diagonal. In next diagonal we have 2 + 9 + 2 = 13 after adding carry 2 we got 15 . Now we underline unit digit 5 and carry the tens digit 1in the next diagonal. (Add shown above with arrows)
Step 4.
Underline all digits obtained in the sum of the topmost diagonal. In topmost diagonal we got 0 + 1 = 1.
Step 5.
Now note the underlined digits, this is your required square number.
Therefore (39)2 = 1521
If you don’t understand read it again and again.
After learning about square, we’ll learn about Square Root .
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Square Root :
If we multiply a number x by itself then we will get a resultant number, then the number x is called the square root of the resultant number.
To represent the square root we use the symbol $$\sqrt{}$$.
$$x\;\times\;x\;=\;x^{2\;\;}\;or\;\;\sqrt{x^2}\;=\;x$$
Example – $$\sqrt{81}\;=\;9$$
$$\sqrt{25}\;=\;5$$
$$\sqrt9\;=\;3$$
Note – Basically square root represent 1/2 power of the number.
Square Root of a perfect square number using the prime factorisation method :
To find the square root of a given perfect square number you have to follow some steps given below –
Step – 1
Find the prime factors of a given number.
Step– 2
Underline the similar factors in pairs.
Step– 3
Choose one factor from each pair and find the product.
324
To understand, some examples are given below.
Example– (1) What is the square root of 324.
Solution – Prime factors of 324 are –
324= 2×2 × 3×3 × 3×3
Now we will take one factor from each pair, and multiply them.
Hence, $$\sqrt{324}\;=\;2\times3\times3\;=\;18$$
Therefore square root of the 324 is 18.
Example –2 Write the square root of 19600
Solution – Prime factors of 19600 are-
19600 = 2×2 × 2×2 × 5×5 × 7×7
Now we will take one factor from each pair and multiply them.
Hence, $$\sqrt{19600}\;=\;2\times2\times5\times7\;=\;140$$
Therefore square root of the 19600 is 140.
After the prime factorisation method, we’ll discuss about Long Division Method to find the square root of a perfect square number.
Long Division Method to find the square root of a perfect square number :
To find the square root of very large numbers, the factorisation method is very large numbers we have to use long division method.
To understand the long division method we have to follow some steps given below.
Step– 1
Make the pairs of digits from the side of the unit digit. Remaining digit ( if any) and every pair is called a period.
Step – 2
Think that the number whose square is equal or just less than the first pair ( period) or remaining digit. You have to take this number as the divisor and quotient also.
Step – 3
Now multiply the divisor and quotient and subtract from the first pair ( period) Now to the right side of the reminder take down the next pair. This is your new dividend.
Step – 4
Now add the quotient with divisor and annex with a suitable hat digit to use as a new divisor. Choose the next digit in such a way the product of the new divisor and this digit should be equal or just less than the new dividend.
Step –5
Repeat step – 2, step-3, and step-4 till all the pairs (period)
have been taken up.
Now the obtained quotient is the required square root.
To understand these steps lets see some examples-
Example – Find the square root of 841
According to first step we make the pairs of digits from the side of the unit digit. Hence first we underline to 41. Since 8 has no pair hence take it single as shown below.
After that think a number whose square is just less than or equal to 8, which is 2. So write 2 in left side and hence up side also.
After subtracting square of 2 (=4) from 8 we have 4 as remainder. Now note down next pair 41 with remainder 4. Multiply divisor 2 with quotient 2 you will get 4 as the tens digit of next divisor. Now think a number of unit place for next divisor and use this unit place digit as quotient . As shown below .
Hence, according to above calculation Square root of 841 is 29.
For better understanding please read it again.
Square root of numbers in decimal form :
As we know that many times we face some mathematical calculation in which we have to find the square root of a number having decimal .
So now we discuss about the method to find the Square root of numbers in decimal form.
Method:
If necessary, make the number of decimal places even by adding zeros. Now, mark the periods and find the square root by long-division method as explained above .
Mark the period from left side after decimal and from right side before decimal.
Place the decimal point in the square root as soon as you finish the integral.
To find the square root which is correct to certain decimal places :
If you want to find the square root of any number, which is correct up to the two decimal places, then you have to find it up to three decimal places and after that round it off up to two decimal places.
Similarly, If you want to find the square root of any number, which is correct up to the three decimal places, then you have to find it up to four decimal places and after that round it off up to three decimal places and so on.
This is all about the Square and Square Roots (Explained with Example). Thanks for visit.
Square and Square Roots (Explained with Example)
Square and Square Roots (Explained with Example)
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