# Square And Square Roots

Hi Students, welcome to **Amans Maths Blogs (AMB)**. In this article, **CBSE Class 8 Maths Chapter Square and Square Roots** is explained. It is the notes of **Square and Square Roots** chapter which help for the students who study in class 8 of CBSE or any affiliated school.

**Square** of a number is the multiplication of the numbers with itself two times.

^{2}

**Square Root** of a number is the number that gives n when it is multiplied by itself.

^{2}= N

For example: Square of 9 is 81 and Square Root of 81 is 9, explained as below.

To check whether a natural number is a perfect square or not, we do the prime factorization of the natural number. If all the primes in prime factorization are in pairs, then it is a perfect square number otherwise not.

Read : Cubes & Cube Roots

For example: 7744 = (2 x 2) x (2 x 2) x (2 x 2) x (11 x 11), here all the primes in the prime factorization are in pairs, so 7744 is a perfect square number.

# Square of Rational Numbers

The square of a rational number (p/q, q ≠ 0) is the ratio of square of its numerator and the square of its denominator.

^{2 }/ q

^{2}

# Properties of Square Numbers

There are following properties of square numbers:

**Property 1:** A number ending with the digits 2, 3, 7 or 8 at unit place is never a perfect square. Conversely, a number ending with 0, 1, 4, 5, 6, 9 may be or may not be perfect square.

For example : 32 is never be a perfect square. 81 is a perfect square but 101 is not a perfect square.

Read : Properties of Integers

**Property 2:** The number of zeroes at the end of a perfect square is always even. It means, a number ending with odd number of zeroes can never be perfect square.

For example : 100, 2500, 36000000 are perfect squares but 1000, 250, 36000 are not a perfect square.

**Property 3:** Square of an even number is always an even number and similarly square of an odd number is always an odd number.

For example : (24)^{2} = 576; here 24 is an even number and its square 576 is also an even number and (17)^{2} = 289; here 17 is an odd number and its square 289 is also an odd number.

**Property 4:** A perfect square leaves remainder 0 or 1 on the division by 3. It means, the square of a natural number is either a multiple of 3 or multiple of 3 plus 1.

Thus. **N ^{2} = 3m or (3m + 1).**

Similarly, a perfect square leaves remainder 0 or 1 or 4 on the division by 5. It means, the square of a natural number is either a multiple of 5 or multiple of 5 plus 1 or multiple of 5 plus 4.

Thus. **N ^{2} = 5m or (5m + 1) or (5m + 4). **

Similarly, **N ^{2} = 7m or (7m + 1) or (7m + 2) or (7m + 4)** and

**N ^{2} = 11m or (11m + 1) or (11m + 3) or (11m + 4) or (11m + 5) or (11m + 9).**

**Property 5:** When a perfect square is divided by the following divisors then the corresponding remainders are obtained as given in table below.

N^{2} is divided by |
Possible Remainders |
---|---|

3 | 0, 1 |

5 | 0, 1, 4 |

7 | 0, 1, 2, 4 |

11 | 0, 1, 3, 4, 5, 9 |

13 | 0, 1, 3, 4, 9, 10, 12 |

**Property 6:** If N^{2} is a perfect square, then 2N^{2} can never be a perfect square. It means, there are no natural numbers p and q satisfy the equation **p ^{2} = 2q^{2}**.

Read : Linear Equations in One Variable

For example: 9 is a perfect square of 3 but 2 x 9 = 18 is not a perfect square of any natural number.

Read : Cubes & Cube Roots

**Property 7:** The sum of first m odd natural numbers is a perfect square and it is equal to the square of m, means m^{2}.

^{2}

For example: 1 + 3 + 5 + 7 + … to 10 terms = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 10^{2} = 100

**Property 8:** The difference of squares of two consecutive natural numbers is equal to their sum. It means, (n + 1)^{2} – n^{2} = (n + 1 + n) (n + 1 – n) = 2n + 1 = (n + 1) + n.

For example: 15^{2} – 14^{2} = 225 – 196 = 29 = 15 + 14

**Property 9:** If the three natural numbers a, b, c (a < b < c) satisfies a relation a^{2} + b^{2} = c^{2}, then (a, b, c) is known as **Pythagorean Triplet** and hence a, b, c are the sides of a right angled triangle. The Pythagorean triplets are in the form of a = 2m, b = m^{2} – 1 and c = m^{2} + 1.

For example: The natural numbers 3, 4, 5 are Pythagorean triplet as 3^{2} + 4^{2} = 5^{2}.

# Pattern of Some Perfect Squares

There are following pattern of perfect square numbers:

**Pattern 1:** The pattern of the squares of 1, 11, 111, 1111 etc as below.

**Pattern 2:** The pattern of the squares of 7, 67, 667, 6667 etc as below.

**Pattern 3:** The pattern of the squares of 9, 99, 999, 9999 etc as below.

**Pattern 4:** The pattern of the squares of 11, 101, 1001, 10001 etc as below.

# Method to Find Squares

There are following method to find the square of a number.

## Column Method To Find Square of a Number

In this method, we use (a + b)^{2} = a^{2} + 2ab + b^{2} to find the square of a natural number having two-digits only.

Let we need to find the square of 79 by using column method.

**Step 1:** Take a = 7 and b = 9, then write the values of a^{2}, 2ab and b^{2} in three columns.

**Step 2:** Now, underline the unit digit of b^{2} and add the tens digit to the column of 2ab.

**Step 3:** Underline the unit digit of the values of 2ab and add then number formed by other digits to the column of a^{2}. Underline its values.

**Step 4:** Write the underlined digits and this is the square of the number ab = 79. Thus, **79 ^{2} = 6241**.

## Diagonal Method To Find Square of a Number

In this method, we can find the square of any natural number having n-digits. To find the square of n-digits natural number, we do the following steps.

**Step 1:** First, we draw a square with n x n (n by n) sub-squares of equal size and draw the diagonals of each sub-squares as shown below.

Here, we are going to find the square of 425 (3-digits natural number), so we draw a 3 x 3 square and sub-squares with diagonal of each sub-square as below.

**Step 2:** Write the digits of the number along the left vertical and the top horizontal sides.

Here, the number is 425. Thus, we write the digits 4, 2, 5 along the left vertical and the top horizontal sides as below.

**Step 3:** Multiply each digit on the left with each on the top one by one write the units digit of the product below the diagonal and tens digit above the diagonal. If the product is one-digit, then make it two-digits by placing 0 before the product.

Here, the digits are 4, 2, 5 then we write the product 4 x 4 = 16, 4 x 2 = 08, 4 x 5 = 20, 2 x 4 = 08, 2 x 2 = 04, 2 x 5 = 10, 5 x 4 = 20, 5 x 2 = 10 and 5 x 5 = 25 as below.

**Step 4:** Starting below the lowest diagonal sum the digits along diagonals obtained. Write the units digit of the sum and take carry, the tens digit (if any) to the diagonal above.

**Step 5:** Write down the digits from the left most side. This is the square of the number. From the above steps, we get the square of 425 as **(425) ^{2} = 180625**.

# Short Trick To Find Square

There are some special natural numbers whose squares can be found by using some short trick methods explained as below. Square and Square Roots.

## Square of Number Ending With Digit 5

Let a two-digit number of the form **A5**, then N = A5 = 10A + 5.

Now, N^{2} = (10A + 5)^{2} = 100A^{2} + 100A + 25 = 100A(A + 1) + 25 = A(A + 1)25, is a four-digit number where A(A + 1) is the product of A and (A + 1 ).

For example: **45 ^{2} = 4(4 + 1)25 = (4×5)25 = 2025**.

## Square of Number Starting With Digit 5

Let a two-digit number of the form **5A**, then N = 5A = 50 + A.

Now, N^{2} = (50 + A)^{2} = 2500 + 100A + A^{2} = 100(25 + A) + A^{2} = (25 + A)A^{2}, is a four-digit number where (25 + A) is the sum of 25 and A.

For example: **57 ^{2} = (25 + 7)(7^{2}) = 3249**.

## Square of Computed Sequentially

If we know the square a natural number A, mean A^{2 }is known, then the square of (A + 1) can be found by trick.

Now, (A + 1)^{2} = A^{2} + 2A + 1 = A^{2} + A + (A + 1)

For example: As we know that **50 ^{2} = 2500** then

**51**.

^{2}= 50^{2}+ 50 + 51 = 2601## Square of Number Between 26 to 50

The square of a natural number between 26 and 50 can be found easily by the trick explained as below.

Now, (25 + A)^{2} – (25 – A)^{2 }= (25 + A + 25 – A)(25 + A – 25 + A) = 50(2A) = 100A.

Thus, (25 + A)^{2}^{ }= 100A + (25 – A)^{2}.

For example: **37 ^{2} = (25 + 12)^{2} = 100 x 12 + (25 – 12)^{2} = 1200 + 169 = 1369**.

## Square of Number Between 51 to 100

The square of a natural number between 51 and 100 can be found easily by the trick explained as below.

Now, (50 + A)^{2} – (50 – A)^{2 }= (50 + A + 50 – A)(50 + A – 50 + A) = 100(2A) = 200A.

Thus, (50 + A)^{2}^{ }= 200A + (50 – A)^{2}.

Or, we can also use the trick as (50 + A)^{2}^{ }= 2500 + 100A + A^{2 }= (25 + A)A^{2}.

For example: **63 ^{2} = (50 + 13)^{2} = 2500 + 1300 + 169 = 3969**

# Square Roots

A number **n** is called the square root of a number **m**, if m = n^{2 }and the square root of m is denoted by **√m**.

If the number of digits in a perfect square is n, then the numerator of the digits in its square root is n/2 (if n is even) or (n + 1)/2 (if n is odd).

# Method to Find Square Roots of Perfect Square

There are following method to find the square root of a number.

## Prime Factorization Method To Find Square Root

To understand this method, we take an example. Let we need to find the square root of 15376. Then, the prime factorization of 15376 is as below.

15376 = 2 x 2 x 2 x 2 x 31 x 31 = (2 x 2) x (2 x 2) x (31 x 31) = (2 x 2 x 31)^{2}.

Thus, **√**15376 = 2 x 2 x 31 = 124

## Long Division Method To Find Square Root

For perfect square, we do the following steps to find the square root of the perfect square.

**Step 1:** Group the digits of the perfect square in pairs, starting with the digit in the units place. Each pair and the remaining digit (if any) is called a period.

**Step 2:** Find the largest number whose square is equal to or just less than the first period. Take this number as the divisor and also as the quotient.

**Step 3:** Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.

**Step 4:** Now, the new divisor is obtained by taking two times the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.

**Step 5:** Repeat the steps 2, 3 and 4 till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number.

For example: Let we need to find the square root of 16384. Using the above steps, we get the square root of 16384 as below.

Thus, **√16384 = 128.**

# Method to Find Square Roots of Non-Perfect Square

To find the square root of the non-perfect square, we use the **Long Division Method**.

**Step 1:** Obtain the number whose square root is to be computed.

**Step 2:** Determine the number of decimal places to which the square root of the number is to be computed. Suppose the square root of the given number is to be computed correct to n places of decimal.

**Step 3:** Count the number of the digits in the decimal part. If the number of the digits is less than **2n**, then affix a suitable number of zeroes at the extreme right of the decimal part so that the number of the digits in decimal part becomes **2n**.

**Step 4:** Use the method of long division to find the square root up to (n + 1) places of decimal.

**Step 5:** Check the digit at (n + 1)th decimal place, if it is less than 5, then delete it to get the square root correct to n decimal places. If the digit at (n + 1)th decimal place is 5 or more than 5, then increase the digit at nth decimal place by one and delete the digit at (n + 1)th place to obtain the square root correct up to n decimal places.

Read : Cubes & Cube Roots

## Square Root of 2

Let we need to find the square root of 2 up to two-decimal places.

The, we need to calculate the square root of 3 up to three decimal places as below.

Thus, **√2 = 1.414. **and the third place is 4 < 5. So the square root of 2 is **√2 = 1.41**. up to two decimal places.

## Square Root of 3

Let we need to find the square root of 3 up to three-decimal places.

The, we need to calculate the square root of 3 up to four decimal places as below.

Thus, **√3 = 1.7320. **and the fourth place is 0 < 5. So the square root of 3 is **√3 = 1.732**. up to three decimal places.

# Important Facts About Square And Square Roots

There are following facts about square and square roots.

**Fact 1:** Square root of a number is always positive. It means, **√16 = 4 NOT -4**.

**Fact 2:** If n^{2} = 16, then n = ±**√**16 = 4 or -4. Both 4 and -4 can be the values of n as the squares of 4 and -4 are 16.

**Fact 3:** **√**a x **√**b = **√**ab is **NOT** valid if both a and b are negative.

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