Square Root – Basics, Formulas, Practice Aptitude Questions and Solved Examples (Prime Factorization, Perfect Squares)

Square Root (Basics, Formulas, Questions and Solved Examples)

What is a Square Root?

Square Root of a number is a value, which on multiplication by itself, gives the original number. The square root is an inverse method of squaring a number. Hence, squares and square roots are related concepts.

Suppose x is the square root of y, then it is represented as x=√y, or we can express the same equation as x 2 = y. Here, ‘√’ is the radical symbol used to represent the root of numbers. The positive number, when multiplied by itself, represents the square of the number . The square root of the square of a positive number gives the original number.

For example, the square of 3 is 9, 32 = 9 and the square root of 9, √9 = 3. Since 9 is a perfect square, hence it is easy to find the square root. But for an imperfect square like 3, 7, 5, etc., we have to use different methods to find the square root.



Square Roots Definition

The square root of any number is equal to a number, which when squared gives the original number.
Let us say m is a positive integer, such that √(m.m) = √(m2) = m

In mathematics, a square root function is defined as a one-to-one function that takes a positive number as an input and returns the square root of the given input number.
f(x) = √x

For example, if x=4, then the function returns the output value as 2.
Note: The square root of a negative number represents a complex number.
Suppose √-n = i√n, where i is the imaginary number.


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Square Root Symbol

The square root symbol is usually denoted as ‘√’. It is called a radical symbol. To represent a number ‘x’ as a square root using this symbol can be written as:
√x
where x is the number. The number under the radical symbol is called the radicand. For example, the square root of 6 is also represented as radical of 6. Both represent the same value.

Square Root Formula

The formula to find the square root is:
y = √a

Since, y.y = y2 = a; where ‘a’ is the square of a number ‘y’.

Properties of Square Root

Some of the important properties of the square root are as follows:
  • If a number is a perfect square number, then there exists a perfect square root.
  • If a number ends with an even number of zeros (0’s), then it can have a square root.
  • The two square root values can be multiplied. For example, √3 can be multiplied by √2, then the result should be √6.
  • When two same square roots are multiplied, then the result should be a radical number. It means that the result is a non-square root number. For instance, when √7 is multiplied by √7, the result obtained is 7.
  • The square root of any negative numbers is not defined because the perfect square cannot be negative.
  • If a number ends with 2, 3, 7 or 8 (in the unit digit), then the perfect square root does not exist.
  • If a number ends with 1, 4, 5, 6 or 9 in the unit digit, then the number may have a perfect square root.

How to find Square Root of Numbers?

To find the square root of any number, we need to figure out whether the given number is a perfect square or an imperfect square. If the number is a perfect square, such as 4, 9, 16, etc., then we can factorize the number by prime factorisation method. If the number is an imperfect square, such as 2, 3, 5, etc., then we have to use a long division method to find the root.

Hence, the methods to find the square root of numbers are:

  • Square Root by Prime Factorisation
  • Square Root by Repeated Subtraction Method
  • Square Root by Long Division Method
  • Square Root by Estimation Method




Square Root by Prime Factorization

The square root of a perfect square number is easy to calculate using the prime factorization method.

Number Prime Factorization Square Root
16 2x2x2x2 √16 = 2×2 = 4
144 2x2x2x2x3x3 √144 = 2x2x3 = 12
169 13×13 √169 = 13
256 256 = 2×2×2×2×2×2×2×2 √256 = (2x2x2x2) = 16
576 576 = 2x2x2x2x2x2x3x3 √576 = 2x2x2x3 = 24

Finding Square Roots by Repeated Subtraction Method

As per the repeated subtraction method, if a number is a perfect square, then we can determine its square root by:
  • Repeatedly subtracting consecutive odd numbers from it
  • Subtract till the difference is zero
  • Number of times we subtract is the required square root

For example, let us find the square root of 25.

  • 25 – 1 = 24
  • 24 – 3 = 21
  • 21 – 5 = 16
  • 16 – 7 = 9
  • 9 – 9 = 0

Since, the subtraction is done for 5 times, hence the square root of 25 is 5.

Finding Square Roots by Long Division Method

Finding square roots for the imperfect numbers is a bit difficult but we can calculate using a long division method. This can be understood with the help of the example given below. Consider an example of finding the square root of 436.


Thus, the square root of 436 is 20.880 (rounding to 3 decimals).

Square Root by Estimation Method

This method is used as an approximation method, to find the square root by guessing the values.
For example, the square root of 4 is 2 and the square root of 9 is 3, thus, we can guess the square root of 5 will lie between 2 and 3.
But, we need to check the value of √5 is nearer to 2 or 3. Let us find the square of 2.2 and 2.8.
  • 2.22 = 4.84
  • 2.82 = 7.84

Since the square of 2.2 gives an approximate value of 5, thus we can estimate the square root of 5 is equal to 2.2 approximately.

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Square Root of Perfect Squares

Below are the numbers which are perfect squares and then finding the square roots of such numbers is easy.
  • 12 = 1 ⇔ √1 = 1
  • 22 = 4 ⇔ √4 = 2
  • 32 = 9 ⇔ √9 = 3
  • 42 = 16 ⇔ √16 = 4
  • 52 = 25 ⇔ √25 = 5
  • 62 = 36 ⇔ √36 = 6
  • 72 = 49 ⇔ √49 = 7
  • 82 = 64 ⇔ √64 = 8
  • 92 = 81 ⇔ √81 = 9
  • 102 = 100 ⇔ √100 = 10

Hence, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 are the "Perfect Squares" here.

√n Value √n Value √n 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 51 7.1414




Square Root of Decimal

A decimal value will have a dot (.) such as 3.8, 5.2, 6.33, etc. For a whole number, we have understood how to derive the square root but let us see how to get the square root of a decimal.

Example: Find the square root of 0.09. Let N2 = 0.09
Taking root on both sides.
N = ±√0.09
As we know,
0.3 x 0.3 = (0.3)2 = 0.09
Therefore,
N = ±√(0.3)2
N = ±(0.3)

Square Root of Negative Number

The square root of a negative number is not a real number but a complex number. It is because the square of any integer is a positive value. The square root of a negative number, say, -y, is: √(-y)= i√y, where ‘i’ is the square root of -1.

How to Solve the Square Root Equation?

A square root equation is such an equation that has a variable in the radicand of the root. It is also called the radical equation.

To solve the radical equation, we need to follow the below steps:

  • Isolate the square root to one of the sides (L.H.S or R.H.S)
  • Square both the sides of the given equation
  • Now solve the rest equation

Example: Solve the radical equation √(4a+9) – 5 = 0
Solution: Given, √(4a+9) – 5 = 0
Isolate the square root term first. Then the equation becomes,
√(4a+9) = 5
Now on squaring both the sides, we get; 4a+9 = 52
4a + 9 = 25
4a = 16
a = 16/4
a = 4

Note: If the equation has more than one radical or square root, then repeat the above given steps for each square root.

Example: Solve √(2x−5) − √(x−1) = 1
Solution: Let us isolate one of the square root.
√(2x−5) = 1 + √(x−1)
Now squaring both the sides
2x – 5 = (1 + √(x−1)) 2
Apply the algebraic identity, (a+b) 2 = a 2 + b 2 + 2ab.
2x−5 = 1 + 2√(x−1) + (x−1)
2x−5 = 2√(x−1) + x
x−5 = 2√(x−1)
Now again isolate the square root.
√(x−1) = (x−5)/2
x−1 = ((x−5)/2) 2
x−1 = (x 2 − 10x + 25)/4
4x−4 = x 2 − 10x + 25
4x − 4 − x 2 + 10x − 25 = 0
−x 2 + 14x − 29 = 0
x 2 − 14x + 29 = 0
Using the quadratic formula, we can solve the above equation.
x = 2.53 and x = 11.47

Applications of Square Roots

The square root formula is an important section of mathematics that deals with many practical applications of mathematics and it also has its applications in other fields such as computing. Some of the applications are:
  • Quadratic Equations
  • Algebra
  • Geometry
  • Calculus




Practice Questions : For Square Roots

Question 1: Solve √10 to 2 decimal places.
Solution:
Step 1: Select any two perfect square roots that you feel your number may fall in between.
We know that 22 = 4; 32 = 9, 42 = 16 and 52 = 25
Now, choose 3 and 4 (as 10 lies between these two numbers)
Step 2: Divide the given number by one of those selected square roots.
Divide 10 by 3.
=> 10/3 = 3.33 (round off answer at 2 places)
Step 3: Find the average of root and the result from the above step i.e.
(3 + 3.33)/2 = 3.1667
Verify: 3.1667 x 3.1667 = 10.0279 (not required)
Repeat step 2 and step 3
Now 10/3.1667 = 3.1579
Average of 3.1667 and 3.1579.
(3.1667+3.1579)/2 = 3.1623
Verify: 3.1623 x 3.1623 = 10.0001 (more accurate)

Question 2: Find the square roots of whole numbers which are perfect squares from 1 to 100.
Solution: The Perfect Squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Square Root Result
1 1
4 2
9 3
16 4
25 5
36 6
49 7
64 8
81 9
100 10

Question 3: What is the square root of 2, 3, 5?
Solution: Use square root list, we have:
√2 = 1.4142
√3 = 1.7321
√5 = 2.2361

Question 4: Is Square Root of a Negative Number a Whole Number?
Solution: No, As per the square root definition, negative numbers shouldn’t have a square root. This is because if we multiply two negative numbers, the result will always be a positive number. Square roots of negative numbers expressed as multiples of i (imaginary numbers).

Question 5: Solve the equation: √(x+2) = 4
Solution: Given, √(x+2) = 4
Squaring both the sides, we get;
x + 2 = 16
x = 14

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