Square Root of a Number: Definition, Formula, and Examples

Just as the division is the inverse operation of multiplication, the square root of a number is the inverse squaring procedure. The square root of any number is multiplied by itself, giving the product equal to the original number.

For example, if \(y\) is the integer and the square root is \(x\), it can be said that \(y×y = x\). The square root is represented by the symbol \(√\). In this article, we will be studying more about the square root of integers, with different examples.

Square Root of a Number

The square root of a given number is that number whose square is the given number. Thus, the square root of a given number \(y\) is the number \(z\), such that \(z \times z\), i.e., \({z^2} = y\).

Hence, if \(z\) is the square of \(y\), then \(y\) is the square root of \(z\). Thus, the symbol \(\sqrt {} \) denotes the positive square root of a number.

Therefore, the square root of \(y\) is denoted as \(\sqrt y \).

For example, \(\sqrt 4 = 2,\sqrt 9 = 3,\sqrt {25} = 5\), etc., and we say the square root of \(4\) equals \(2\). The square root of \(9\) equals \(3\). The square root of \(25\) equals \(5\) and so on.

Square Root of a Number Properties

1. A perfect square number has a perfect square root.

2. If the last digit of a number is even, then its square root will also be even.

3. The odd perfect number has a square root that is odd.

4. The square root of a negative number is not defined or undefined because the perfect square can never be negative.

5. If the number of digits \(x\) in the given number is even, then the number of digits in its square root is \(\frac{x}{2}\).

6. If the number of digits \(x\) in the given number is odd, then the number of digits in its square root is \(\frac{{x + 1}}{2}\).

Facts of Square Root of a Number

1. \(5 \times 5 = 25\) and \(( – 5) \times ( – 5) = 25\). Hence, squares of both \(5\) and \({\rm{( – 5)}}\) is \(25\). Therefore, by definition, the square root of \(25\) is \(5\) and \({\rm{( – 5)}}\). Thus, there are \(2\) square roots of a positive number, out of which one is a positive number, and the other is a negative number.

2. The square root of the product of two or more positive numbers is equal to the product of their square roots. For example, \(\sqrt 5 \times \sqrt 4 = \sqrt {20} \)

3. The square root of an even perfect number is an even number—for example, \(\sqrt {16} = 4,\sqrt {36} = 6\) etc.

4. The square root of an odd perfect number is an odd number—for example, \(\sqrt 9 = 3,\sqrt {49} = 7\) etc.

5. If the number of zeroes at the end of the perfect square number is even, then the square root of that number is an integer. For example, \(\sqrt {100} = 10,\sqrt {400} = 20\).

How do you Calculate the Square Root of a Number?

Before finding the square root of any number, we need to figure out whether the number is a perfect square or an imperfect square. If the number is a perfect square, such as \({\rm{4,9,16,25,36,49}}\) etc., then with the help of the prime factorization method, we can factorize and thus find the square root. But, if the number is an imperfect square, such as \({\rm{2,3,5,7,8}}\) etc., then we have to apply the long division method.

Let us discuss each way of finding the square root one by one with the help of examples.

Square Root of a Number Using Prime Factorization Method

Suppose we need to find the square root of a number. The prime factorization method means that we break the number into all its prime factors and then use those prime factors to find the square root.

This is explained below with the help of an example.

Example: Find the square root of \(576\) by the prime factorization method.

Thus, prime factors of \(576\) are

\(576 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\)

\( = 4 \times 4 \times {2^2} \times {2^2} \times {3^2}\)

\( = {4^2} \times {2^2} \times {3^2}\)

\( = {(4 \times 2 \times 3)^2}\)

\( = {24^2}\)

Therefore, \(\sqrt {576} = \sqrt {{{24}^2}} = 24\)

Hence, the square root of \(576\) is \(24\).

Square Root of a Number Using Long Division Method

Finding the square root of a large number using the prime factorization method is quite time-consuming. Thus, we use another method known as the long division method to find the square root of large numbers.

This process involves several steps and is explained below with the help of an example.

Example: Find the square root of \(7569\) using the division method.

Step 1: Firstly, put a bar over every pair of digits in the given number, starting from the unit’s place. Thus, for \(7569\), we will refer to \(75\) as the first group and \(69\) as the second group.

Step 2: Start with the first group, i.e., \(75\). Think of the nearest perfect square number equal to or less than \(75\)

\( \Rightarrow 64,\) or \({8^2} < 75\)

Therefore, \(8\) becomes the divisor as well as the first digit of the quotient. Carry out the long division to get \(11\) as the remainder.

Step 3: Bring down the second group \({\rm{ (69) }}\);next to the remainder \({\rm{ (11) }}\) to get the new dividend, i.e., \(1169\).

Step 4: Double the previous divisor \({\rm{ (8) }}\) to get the hundreds and tens place digit of the new divisor, i.e., \(16\). To determine the units place digit of the new divisor, which also becomes the units place digit of the final quotient, the digit should be such that the new divisor multiplied by this number will give the new dividend \({\rm{(1169)}}\).

Given that \(1169\) ends with \(9\), the last digit of the new divisor will be \(7\) as \(7 \times 7 = 49\). Let us verify this:

\(167 \times 7 = 1169\), which gives us the required units place digit as \(7\). Therefore, we put \(7\) in the units place of the final quotient and get \(167\) as the new divisor.

Step 5: Carry out the long division, and thus the remainder will be \(0\).Therefore, the square root of \(7569\) is \(87\).

Square Root of a Decimal Number

A similar process can be followed to find the square root of a decimal number. The difference lies in how the numbers are grouped in the first step. Let us understand to find the square root of a decimal number with the help of an example.

Example: Find the square root of \({\rm{38}}{\rm{.44}}\).

Step 1: Put a bar over every pair of digits in the given number, starting from the unit’s place and moving left for an integral part. Further, start with the tenth place and move right for the decimal part. Thus, \(38\), forms a group for an integral part, and \(44\) forms the group for the decimal part.

Step 2: Starting with the first group, i.e., \(38\), think of the nearest perfect square number equal to or less than \(38\), which is \(36 = {6^2}\). Therefore, \(6\) becomes the divisor, as well as the first digit of the quotient. Carry out the long division to get \(2\) as the remainder.

Step 3: Bring down the second group, i.e., \(\left( {44} \right)\) next to the remainder \(2\), to get the new dividend \({\rm{(244)}}\). Since the decimal part has come down, place the decimal point in the quotient.

Step 4: Double the previous divisor \({\rm{(6)}}\) to get the hundreds and tens place digit to the new divisor, i.e., \(12\). To determine the units place digit of the new divisor, which now becomes the decimal part of the final quotient, the digit should be such that the new divisor multiplied by this number will give the new dividend \({\rm{(244)}}\).

Given that \(244\) ends with \(4\), the last digit of the new divisor can be either \(2\) or \(8\). If it is \(2\),

\(122 \times 2 = 244\), which gives us the required units, place digit as \(2\). Therefore, we put \(2\) in the decimal part of the final quotient and get \(122\) as the new divisor.

Step 5: Finally, carry out the long division, and thus the remainder will be \(0\). Therefore, the square root of \({\rm{38}}{\rm{.44}}\) is \({\rm{6}}{\rm{.2}}\).

Estimating the Square Root of a Number

Consider a real-life situation:

Vaibhav has square plywood that has an area of \(200\;{\rm{c}}{{\rm{m}}^2}\). He wants to use the wood as a backing for the square mirror. Mirror frames come in whole numbers unit lengths. What is the largest length of frame that Vaibhav can buy to make the most use of his piece of plywood?

Since we know that the length required is a whole number, and the square of the length should come close to \(200\;{\rm{c}}{{\rm{m}}^2}\), we can estimate the square root of \(200\;{\rm{c}}{{\rm{m}}^2}\) without calculating the actual square root.

We know that, \({14^2} = 196\) and \({15^2} = 225\). Since \({\rm{196 < 200 < 225}}\), and \(200\) is closer to \(196\) (or \({14^2}\)), we can estimate that \(14\;{\rm{cm}}\) will be the length of the largest frame that Vaibhav can buy to make the most use of his plywood of \(200\;{\rm{c}}{{\rm{m}}^2}\).

In mathematical terms, \(\sqrt {200} \approx 14\) read as the square root of \(200\) is approximately equal to \(14\).

Solved Examples

Q.1. The area of a square is \(1,521\,{\rm{c}}{{\rm{m}}^2}\). Find the length of a side of the square by the prime factorization method.
Ans: Area of a square \( = {\rm{ length }} \times {\rm{ breadth }}\)

Therefore, the length of the square \( = \sqrt {{\rm{ area }}} \)

Now, \(1521 = 3 \times 3 \times 13 \times 13 = {3^2} \times {13^2}\)

\(\sqrt {200} = \sqrt {{3^2}} \times \sqrt {{{13}^2}} \)

\( = 3 \times 13 = 39\)

Hence, the length of a side of a square is \(39\;{\rm{cm }}\).

Q.2. Find the square root of \(841\) using the division method.
Ans: Let us do the division as shown below.

Hence, the square root of \(841\) is \(29\).

Q.3. Find the square root of \(36\) by the prime factorization method.
Ans: \(36 = 2 \times 2 \times 3 \times 3 = {2^2} \times {3^2}\)

Therefore, \(\sqrt {36} = \sqrt {{2^2}} \times \sqrt {{3^2}} = 2 \times 3 = 6\)

Q.4. Find the square root of \(2\) correct up to three decimal places.
Ans: We will find the square root of \(2\) by the division method.

Thus, the square root of \({\rm{2 = 1}}{\rm{.414}}\).

Q.5. Find the square root of \(144\) by the prime factorization method.
Ans: The prime factors of \(144\) are:

\(144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = {2^4} \times {3^2}\)

\(\sqrt {144} = \sqrt {{2^4}} \times \sqrt {{3^2}} = {2^2} \times 3\)

\(\sqrt {144} = 12\)

Hence, the square root of \(144\) is \(12\).

FAQs

Q.1. How do you calculate the square root of a number?
Ans: Square root of a number can be calculated by either the prime factorization method or the long division method.

Q.2. Explain the square root of a number with an example?
Ans: The square root of a given number is that number whose square is the given number. Thus, the square root of a given number \(y\) is the number \(z\), such that \(z \times z\), i.e., \({z^2} = y \cdot \sqrt 4 = 2,\sqrt 9 = 3,\sqrt {25} = 5\), etc., and we say the square root of \(4\) equals \(2\). The square root of \(9\) equals \(3\).

Q.3. How do you find the square root of an imperfect square?
Ans: The square root of an imperfect square can find out by the long division method.

Q.4. What is the square of \(19\)?
Ans: The square of \(19\) is \(361\).

Q.5. How do you find the square root of a perfect square?
Ans: The square root of perfect squares can find out by using prime factorization.

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