Squares and Square Roots: Definition, Calculation, Examples

Squares and Square Roots: Squares and square roots are inverse operations in mathematics. The square root of a number is the number that needs to be multiplied by itself to get the original number, whereas the square of a number is the number that needs to be multiplied by itself to get the actual number. If \(p\) is the square root of \(r,\) then \(p×p=r\)  is true. Every number has two square roots, one positive value and one negative value. Let us discuss Squares and Square roots in detail.

Squares and Square Roots: Definition

The square of a number is the number multiplied by itself.
\(2 \times 2 = {2^2} = 4,3 \times 3 = {3^2} = 9,4 \times 4 = {4^2} = 16\)
\(4\) is the square of \(2,  9\) is the square of \(3, 16\) is the square of \(4\)

In these examples, \(4, 9\) and \(16\) are natural numbers, which are the squares of natural numbers, \(2, 3,\) and \(4.\,4, 9\) and \(16\) are known as perfect squares.

Perfect Square

A perfect square or a square number is a natural number which is the square of another natural number. Consider the numbers \(132\) and \(225.\) These can be written as a product of their prime factors as
\(132=2×2×3×11\)
\(225=3×3×5×5 \)    

The number \(132\) does not have pairs of identical factors. Thus, \(132\) cannot be a perfect square. The number \(225,\) however, has pairs of identical factors.
\(225 = 3 \times 3 \times 5 \times 5 = {3^2} \times {5^2} = {(3 \times 5)^2} = {15^2}\)

Therefore \(225\) is a perfect square. It is the square of \(3×5\) or \(15.\)

Properties of Squares

1. A number ending in \(2, 3, 7\) or \(8\) is never a perfect square.
Thus, \(32532, 4353, 5007, 384898\) are not perfect squares.
The squares of the first \(10\) natural numbers are given table.

2. A number ending in an odd number of zeros is never a perfect square.
Examples: \(3530, 55849000, 38485000\) are not perfect squares.
However, this does not mean that all numbers ending in an even number of zeros are always perfect squares. These may or may not be perfect squares.
Examples:
\(500\) is not a perfect square.
\(400\) is a perfect square.

3. The square of even numbers is always even.
The square of a natural number \(n\) is equal to the sum of the first \(n\) odd numbers. Thus, \({1^2} = 1\)
\({2^2} = 1 + 3;\) a sum of the first \(2\) odd numbers.
\({3^2} = 1 + 3 + 5;\) a sum of the first \(3\) odd numbers.

4. The difference of the squares of two consecutive numbers is equal to the sum of the numbers.
Example: \({11^2} – {10^2} = 121 – 100 = 21 = 11 + 10\)

5. Three natural numbers \(a, b\) and c are called a Pythagorean triplet if \({a^2} + {b^2} = {c^2}\)
For any number \(n > 1,\left( {2n,{n^2} – 1,{n^2} + 1} \right)\) is a Pythagorean triplet.    
Example: for \(n = 3,(6,8,10)\) is a Pythagorean triplet.
Since, \({10^2} = 100 = {6^2} + {8^2}\)

Square Roots

A number’s square root is the number that is multiplied by itself to produce the product. Exponents are something we have learned about. Special exponents include squares and square roots. Think about the number nine. When \(3\) is multiplied by itself, the result is \(9.\)

A square is a number with an exponent of two. It is called a square root when the exponent is \(\frac{1}{2}.\) For instance, \((n \times n) = \sqrt {{n^2}} = n,\) where \(n\) is a positive integer.

Square Root Definition

The square root of a number \(n\) is that number which, when multiplied by itself, gives \(n.\) The square root of \(n\) is denoted as \(\sqrt n .\) The symbol \(\sqrt {} \) It means ‘square root of’.

Examples: \(\sqrt {16} = 4,\sqrt {25} = 5,\sqrt {169} = 13\) 

Square Root of Perfect Squares

1. Prime Factorisation Method:
The prime factors of a perfect square can be paired. To find the square root, we first make pairs of the pairs of the prime factors. We then take one factor of every pair and find the product of these factors. The method helps find the square root of the perfect square.

Example: Find the square root of \(4356.\)

Factorizing \(4356\) by the division method, we can write \(4356\) as a product of its prime factors as
\(4356=2×2×3×3×11×11\)
\(\sqrt {4356} = 2 \times 3 \times 11 = 66\)

So, the square root of \(4356\) is \(66.\)

2. Pattern Method for Square Root of a Perfect Square Ending in \(25:\)

Let us find the square root of \(7225.\)

1. As the perfect square ends in \(25,\) its square root must end in \(5.\) Write \(5\) in the units place.
2. Find two single-digit number numbers whose product is \(72.\) So, we have \(8×9=72.\) Now we need to consider the smaller number between those two numbers \(8\) and \(9.\) In this case, it is \(8.\)
3. \(8\) must be present in the tens; hence \(\sqrt {7225} = 85\)

Long Division Method

The prime factorisation method becomes difficult for large numbers. In such cases, we use the long division method, which consists of two steps:

1. Pairing
2. Long division

The pairing of numbers in a decimal number to find square roots by the long division method is done from right to left of the whole number portion and from left to right for the decimal part.

Example: Find the square root of \(6889\)

Square Root of Fractions

Example: Find the square root of \(\frac{{384}}{{1014}}.\)

Here the numerator and the denominator are not perfect squares. However, if they are both divided by the common factor \(6,\) they become perfect squares.

Thus,
\( \Rightarrow \frac{{384}}{{1014}} = \frac{{64}}{{169}}\)
\( \Rightarrow \sqrt {\frac{{384}}{{1014}}} = \sqrt {\frac{{64}}{{169}}} = \frac{{\sqrt {64} }}{{\sqrt {169} }} = \frac{8}{{13}}\)

Square Roots of Decimal Numbers

While calculating the square root of a decimal number, remember that the square root will have half the number of decimal places as the number itself has.

Examples:

1. \(\sqrt {0.01} = 0.1\) since \(0.1 \times 0.1 = 0.01\)
2. \(\sqrt {0.64} = 0.8\) since \(0.8 \times 0.8 = 0.64\)

In each case, the number has two digits in the decimal part, while the square root has only one.

Square Root of Decimal by Long Division Method

Example: Find the square root of \(7624.7824\)

Steps to find the Square Root

1. Pair off the numbers before the decimal point from right to left. The periods are \(76\) and \(24.\)
2. Since the number of decimal places is even, there is no need to put \(0.\) Pair off the numbers after the decimal point from left to right. The periods are \(78\) and \(24.\)
3. Compute the square root by the long division method as shown.

\(\sqrt {7624.7824} = 87.32\)

Solved Examples – Squares and Square Roots

Let us look at some of the solved examples about squares and square root:

Q.1. Find the square root of \(\frac{{36}}{{49}}.\)
Ans: The given fraction is \(\frac{{36}}{{49}}\)
The square root of \(36 = \sqrt {36} = 6\)
The square root of \(49 = \sqrt {49} = 7\)
So, the square root of \(\frac{{36}}{{49}} = \sqrt {\frac{{36}}{{49}}} = \frac{6}{7}\)
Therefore, \(\sqrt {\frac{{36}}{{49}}} = \frac{6}{7}.\)

Q.2. Find the value of \(\sqrt {25} \times \sqrt {441} .\)
Ans: The given expression is \(\sqrt {25} \times \sqrt {441} \)
The square root of \(25 = \sqrt {25} = 5\)
The square root of \(441 = \sqrt {441} = 21\)
\( \Rightarrow \sqrt {25} \times \sqrt {441} \)
\(=5×21=105\)
Therefore, the value of \(\sqrt {25} \times \sqrt {441} = 105\)

Q.3. What is the difference of \({22^2}\) and \({21^2}.\)
Ans: In this case, we need to find \({22^2} – {21^2}.\)
Square of \(22 = {22^2} = 22 \times 22 = 484\)
Square of \(21 = {21^2} = 21 \times 21 = 441\)
\( \Rightarrow {22^2} – {21^2}\)
\( = 484 – 441 = 22 + 21 = 43\)
Therefore, the difference of \({22^2}\) and \({21^2}\) is \(43.\)

Q.4. Find the least number by which \(52\) should be multiplied to make it a perfect square.
Ans: By prime factorization, we get \(52=2×2×13\)
To make \(52\) a perfect square, the number should have pairs of prime factors.
Therefore, multiplication by \(13\) is necessary.
Thus, \(52×13=2×2×13×13\) is a perfect square.

Q.5. Find the square root of \(0.0121.\)
Ans: The square root of \(0.0121\) will have \(2\) decimal places.
The square root of \(121\) is \(11.\)
Therefore, the square root of \(0.0121=0.11.\)

Q.6. Check if \(58564\) is a perfect square.
Ans: Factorizing \(58564\) by the division method we get

\(58564=2×2×11×11×11×11\)
Therefore, \(58564\) is a perfect square.

It is the square of \(2×11×11=242.\)

Summary About Squares and Square Roots

The square of a number is the number multiplied by itself. The square root is the number that equals that value when multiplied by itself. In this article, we studied the definitions of squares and square roots, different methods to find the square root of numbers and decimal numbers with examples. 

It helps for a good understanding of squares and square roots. The outcome of this article helps in solving the different problems on squares and square roots easily.

How To Find Square Root By Long Division Method?

FAQs About Squares and Square Roots

Let us look at some of the frequently asked questions below:

Q.1. How do you solve square roots with squares?
Ans: We can solve square roots with squares by using two methods:
1. Prime Factorizing method
2. Long Division Method

Q.2. What is the relationship between squares and square roots?
Ans: Both square roots and squares are concepts that are opposite in nature. Squares are the numbers that are produced when a value is multiplied by itself. The square root of a number, on the other hand, is a value that, when multiplied by itself, returns the original value.

Q.3. What are perfect squares and square roots?
Ans: The square root is the number that equals that value when multiplied by itself. The square root of \(9\) is \(3\) because \(3\) multiplied by itself is \(9.\) The number four is a perfect square. A value with a whole number square root is called a perfect square.

Q.4. Why is it important to learn the squares and square roots of the numbers?
Ans: A square root is a number that equals that value when multiplied by itself. Because \(5\) times itself equals \(25,\) the square root of \(25\) is \(5.\) A perfect square is made up of four numbers. A value with a whole number square root is referred to as a perfect square.

Q.5. What are the roots of perfect squares?
Ans: Squaring is the process of multiplying a number by itself. Perfect square numbers are those whose square roots are whole numbers (or, more precisely, positive integers).

Now you are provided with all the necessary information on squares and square roots and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.