Squares and Square Roots

Introduction to Square Roots: Definition & Long Division Method

A square root is a value that is obtained when the original number is multiplied by itself. When a number multiplies itself, the product that is obtained is known as the square number. The number is the square root. We get perfect square roots for a perfect square number. A square root is represented by a √ sign. If x2 is a square number then x is a square root of it. Students might still be confused and would want to know – what is square root? Let us look at some of the key concepts including square root definition, ways to figure out the square of a number, square root table, and the square root of numbers.

Square Root Definition

The square root of any number is equal to a number, which when multiplied with the same number gives the original number. We get perfect square roots for a perfect square number. A square root is represented by a √ sign. If x2 is a square number then x is a square root of it.

Consider 12 = 1, therefore square root of 1 is 1. 22 = 4, therefore square root of 4 is 2. Similarly, 92 = 81, therefore the square root of 81 is 9. It is also interesting to know that 92, gives 81, and −92 also gives 81.

Methods to Find Square Root of Numbers

To find out the square root of numbers, it is necessary to first figure out whether the number is a perfect square or an imperfect square. A perfect square is defined as a number that can be expressed as the square of the number from the same number system. On the other hand, imperfect squares are those numbers whose square roots contain fractions or decimals.

If the number turns out to be a perfect square, then the number can be factorized using the prime factorization method. Perfect squares could be 4, 9, 16, 25, etc. However, in the event that the number is an imperfect square, then the square root by long division method will be used to find out the square root of a number. Some examples of an imperfect square are 2, 3, 5, etc.

Some of the key methods to find out the square root of a number are as follows:

  1. Square root by Repeated Subtraction Method
  2. Square root by Prime Factorization
  3. Square root by Estimation Method
  4. Square root by Long Division Method

What is Square root by Repeated Subtraction Method?

The repeated subtraction method is one of the commonly used methods to find out the square root of a number. In this method, the perfect square number is repeatedly subtracted with successive odd numbers, i.e. 3,5,7,9, etc. till we get zero as the remainder. The subtraction begins from 1 and goes on to 3, 5, 7, etc. until zero is derived. This method involves the number of times the value is subtracted to attain zero. This count denotes the square root of numbers that is desired.

25 – 1 24
24-3 21
21-5 16
16-7 9
9-9 0

 

As is seen in the above table, 5 instances of subtraction are being done to achieve zero as the remainder. The subtraction begins with 1 and goes on until the odd number 9. To sum it up, 1,3,5,7, and 9 are subtracted. This accounts to 5 instances. Thus, 5 is the square root of 25.

What is Square Root by Prime Factorization Method?

To achieve the square root of numbers, the prime factorization method is easily used. In this method, the perfect square is factorized into its prime factors by dividing it successively. The pairs of the prime factors are then paired. When we take the product of one factor from each pair, it will result in the square root of the perfect square.

Let us find the square root of 144.

The prime factorization of 144 = 2 × 2 × 2 × 2 × 3 × 3.

When we pair the prime factors and select one from each pair, we have 2 × 2 × 3 = 12. Hence, the square root of 144 is 12.

What is Square Root by Estimation Method?

The square root by estimation method is used as an approximation method. In this, the square root of numbers is figured out by guessing the values. For eg, the square root of 4 is 2, while the square root of 9 is 3. Thus it is easy to figure out that the square root of 5 will be between 2 and 3.

However, we will still have to check the value of √5 is nearer to 2 or 3. Let us try finding out the square of 2.2 and 2.8.

The square of 2.2 = 4.84

The square of 2.8 = 7.84

Since the square of 2.2 is 4.84, which is approximately 5, we can say that the square root of 5 is approximately equal to 2.2.

What is Square Root by Long Division Method?

It is difficult to find out the square root of numbers that are imperfect squares. However, this can be easily done with the use of the long division method. Let us see the steps to find out the square root by long division method by finding out the square root of 225. First, place a bar over the pair of digits in 225, starting from the unit place.

The division should begin from the left mode side of the number. Here 1 is the number whose square is less than 2. Putting it in the divisor and the quotient and then doubling it will give the following result.

Now, we are supposed to find a number for the blanks in divisor and quotient. Let this number be x. We need to check when 2x multiplies by x gives a number that is either less than or equal to 125. Take x = 1, 2, 3, and so on and check.

In this case, 24 × 4 = 96 and 25 × 5 = 125. So we choose x = 5 as the new digit to be put in the divisor and in the quotient. The remainder here is 0 and hence 15 is the square root of 225.

Properties of Square Root

  • A perfect square root exists for a perfect square number only.
  • The square root of an even perfect square is even.
  • An odd perfect square will have an odd square root.
  • A perfect square cannot be negative and hence the square root of a negative number is not defined.
  • Numbers ending with (having unit’s digit) 1, 4, 5, 6, or 9 will have a square root.
  • If the unit digit of a number is 2, 3, 7, or 8 then a perfect square root is not possible.
  • If a number ends with an odd number of zeros, then it cannot have a square root. A square root is only possible for an even number of zeros.
  • Two square roots can be multiplied. √5, when multiplied by √2, gives √10 as a result.
  • Two same square roots are multiplied to give a non-square root number. When √25 is multiplied by √25 we get 25 as a result.

Real-Life Application of Square Roots

Square roots are used in almost every known field. Its most common application relates to the calculation of sides of a square when its area is known. Other key applications that every student should know are:

  • To calculate the length of the diagonal of a rectangle or a square.
  • In Pythagoras theorem to calculate the third side when any of the two sides of a right-angled triangle is known.
  • To calculate the standard deviation from the variation of the data set.
  • To solve quadratic equations.

What is Square Root Table?

After knowing what is square root and what the square root definition is, let us look at the square root table. To help students have a better understanding of the topic, here is a well-prepared square root table. This square root table includes the square root of numbers 1 to 50, which consist of both – perfect square values and imperfect square values.

√number Value √number Value √number Value
√1 1 √18 4.2426 √35 5.9161
√2 1.4142 √19 4.3589 √36 6
√3 1.7321 √20 4.4721 √37 6.0828
√4 2 √21 4.5826 √38 6.1644
√5 2.2361 √22 4.6904 √39 6.2450
√6 2.4495 √23 4.7958 √40 6.3246
√7 2.6458 √24 4.8990 √41 6.4031
√8 2.8284 √25 5 √42 6.4807
√9 3 √26 5.0990 √43 6.5574
√10 3.1623 √27 5.1962 √44 6.6332
√11 3.3166 √28 5.2915 √45 6.7082
√12 3.4641 √29 5.3852 √46 6.7823
√13 3.6056 √30 5.4772 √47 6.8557
√14 3.7417 √31 5.5678 √48 6.9282
√15 3.8730 √32 5.6569 √49 7
√16 4 √33 5.7446 √50 7.0711
√17 4.1231 √34 5.8310    

What is Square Root Formula?

The square root formula is used to find out the square root of numbers. To simplify, the square root formula is, y=√x

It is important to note that y.y=x. Here x is the square of a number y.

For eg, 2= √4, where y=2 and √x=4, thus y.y=x, i.e. 2×2=4.

Square Root of a Negative Number

It is necessary to understand that negative numbers also have square roots. However, the square root of numbers that are negative, are not real numbers but complex numbers. This is because the square of any integer is a positive number. For eg, the principal square root of a number “-x” = √(-x)= i√x. In this, “i” is the square root of -1.

Let us have a look at another example. We consider the square root of a perfect square number like 16. Now let us consider the square root of -16. There is no real square root of the number -16.

√(-16)= √16 × √(-1) = 4i (as, √(-1)= i)

Here, “i” is represented as the square root of -1. Hence, 4i is the square root of the number 16.

How To Find Square Of A Number?

Any number that is raised to the exponent of 2 (y2) is called the square of the base of that number. Hence, 5is referred to as the square of number 5, while 82 is referred to as the square of number 8.

It is easy to find out the square of a number, simply by multiplying the number with the same number once. For example, 5 should be multiplied by 5 to find out the square of the same number. Thus 5×5=25, where 25 is the square of 5. Similarly, 8 should be multiplied by 8 to find out the square of the same number. Thus, 8×8 = 64, where 64 is the square of the number 8.

When we calculate the square of a whole number, the derived number is a perfect square. Some examples of perfect squares are 4, 9, 16, 25, 36, etc. The square of a number, irrespective of being positive or negative, is always a positive number.

Frequently Asked Question

Q: What is the Square Root definition?

  • The square root of any number is equal to a number, which when multiplied with the same number gives the original number. We get perfect square roots for a perfect square number. A square root is represented by a √ sign. If x2 is a square number then x is a square root of it.

Q: What do the terms squares and square roots refer to?

  • When a number is multiplied by the same number once, the product that is obtained is called a square. For instance, the square of 5 (52) can be obtained through multiplication, i.e. 5×5=25. Thus, the square of 5 is 25
  • Meanwhile, the concept of square roots is completely opposite. The square root of any number is equal to a number, which when multiplied with the same number gives the original number. From the above example, it can be ascertained that 25 is the square root of 5.

Q: What are the key methods in obtaining the square root of numbers?

  • There are four key major methods to obtain the square root of numbers and should be known to every student:
  • Square Root by Subtraction Method
  • Square Root by Prime Factorization Method
  • Square Root by Estimation Method
  • Square Root by Long Division Method

Q: Which method helps in finding the square root of imperfect squares?

  • The square root of imperfect squares can be ascertained by using the Long Division Method. The following steps are helpful in finding out the imperfect square:
  • Take the number whose square root is to find.
  • Place a bar over every pair of the digit of the number starting from that in the unit’s place (rightmost side).
  • We divide the leftmost number by the largest number whose square is less than or equal to the number under the leftmost bar.
  • Take this number as the divisor and the quotient. The number under the leftmost bar is considered to be the dividend.
  • Divide and get the number.
  • Bring down the number under the next bar to the right of the remainder.
  • Double the divisor (or add divisor to itself).
  • To the right of this divisor find a suitable number which together with the divisor forms a new divisor for the new dividend. The new number in the quotient will have the same number as selected in the divisor. The condition is the same as being either less or equal to that of the dividend.

Q: What does this symbol √ denote?

  • The symbol √ is used to denote the square root of a number. For example, to find the square root of the number 25, we write it as √ 25= 5.

Q: What are the key properties of square root?

  • A perfect square root exists for a perfect square number only.
  • The square root of an even perfect square is even.
  • An odd perfect square will have an odd square root.
  • A perfect square cannot be negative and hence the square root of a negative number is not defined.
  • Numbers ending with (having unit’s digit) 1, 4, 5, 6, or 9 will have a square root.
  • If the unit digit of a number is 2, 3, 7, or 8 then a perfect square root is not possible.
  • If a number ends with an odd number of zeros, then it cannot have a square root. A square root is only possible for an even number of zeros.
  • Two square roots can be multiplied. √5, when multiplied by √2, gives √10 as a result.
  • Two same square roots are multiplied to give a non-square root number. When √25 is multiplied by √25 we get 25 as a result.

 

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