What is Square Root?
The square root is the opposite operation of squaring. When we find \(5^2\) we multiply 5\(\times\)5 to get 25. On the other hand, if we want to find the square root of 25, we need to factorise it. 25 can be represented as 5\(\times\)5.
Therefore, the square root of 25 is 5. The square root is represented by the symbol ” \(\sqrt{}\) “, known as a radical. The number placed inside the radical is referred to as a radicand. The square root of perfect square numbers is a rational number. The square roots of imperfect squares are irrational. This suggests that their exact values can never be determined. For instance, \(\sqrt{3}\) is irrational and its value is 1.7320508075…
How can we calculate the square root of a number?
We can calculate the square root of any given number that is the square of a positive or negative integer. Let us say we need to calculate the square root of 9, but 9 is the square of 3, and we know that 9 = 3\(\times\)3.
In this case, the number in the radical is a perfect square. However, we must apply different approaches to find the square root of imperfect squares. Long division is one of the methods. For larger numbers, there is a basic method that can be used to obtain the square root, which will be discussed in the next section.
The factor tree
Let us look at a tree diagram and how we can get the root of a number. The factor tree is used to simplify a larger number as a product of its prime factors. Take the example of the number given below.
1764 = 2\(\times\)882
882 = 2\(\times\)441
441 = 3\(\times\)147
147 = 3\(\times\)49
49 = 7\(\times\)7
Each number is broken down as a product of a relatively smaller number and its prime factor. Two of the same prime factors are coupled together as one, and they are multiplied together to give us the square root of the larger number. In this case, we have the factors 2, 3, and 7 occurring twice. Therefore, the number 1764 can be written as:
\(2^2\times3^2\times7^2=1764\)
Now, to find the square root, we can take the square roots of all the factors shown above,
\(\sqrt{2^2}\times\sqrt{3^2}\times\sqrt{7^2}=2\times3\times7\)
Therefore, the square root of 1764 is,
\(2\times3\times7=42\)
So, 42 is the square root of 1764.
Solved Square Root of a Number Examples
Example 1:
Find the length of one side of the square when the area of the square is given.
1)
The length of the square can be found by finding the square root of the area of the square.
Let A be the area of the square and x be the side of the square.
A = \(x^2\)
x = \(\sqrt{A}\)
x = \(\sqrt{49}\)
7 times 7 is 49.
So the square root of 49 is 7.
x = 7
2)
Let A be the area of the square and x be the side of the square.
A = \(x^2\)
x = \(\sqrt{A}\)
x = \(\sqrt{\frac{81}{25}}\)
9 times 9 is 81. So the square root of 81 is 9.
5 times 5 is 25. So the square root of 25 is 5.
Therefore,
\(\sqrt{\frac{81}{25}}=\frac{\sqrt{81}}{\sqrt{25}}\)
= \(\frac{9}{5}\) inches
Therefore the length of the side is \(\frac{9}{5}\) or 1.8 inches.
Example 2:
Find the square root of the following numbers
A. \(\sqrt{49}\)
B. – \(\sqrt{121}\)
C. ± \(\sqrt{25}\)
Answers:
1. Seven times seven gives us 49, so the square root of 49 is 7. \(\sqrt{49}\) represents the positive square root.
\(\sqrt{49}\) = 7
2. Eleven times eleven gives us 121, so the square root of 121 is 11. Since there is a negative sign outside the radical, – \(\sqrt{121}\) represents a negative square root. So the final answer will be – 11.
– \(\sqrt{121}\) = – 11
3. Five times five gives us 25, so the square root of 25 is 5. Since there is a ‘±’ sign outside the radical, ± \(\sqrt{25}\) represents both the positive and negative square roots.
Example 3:
Find the square root of 625.
Answer:
The factor tree of 625 is as follows.
625 = 125\(\times\)5
125 = 25\(\times\)5
25 = 5\(\times\)5
After grouping the prime factors, we get 5\(\times\)5 = 25. So the square root of 625 is 25.
Rapid recall
Given below are the square roots of a few numbers. Memorizing these values will come in handy when working through different questions.
One of the ways to find the square root of a number is by using the factor tree. The factor tree is a method in which the given number is split into a relatively smaller number and a prime factor. The repeated prime factors are grouped together. Each of these grouped numbers is multiplied together to get the square root of the larger number.
We can find the square root of a number by using long division as well.
A square of a negative number is always positive, for instance,
\(-2^2=(-2)\times(-2)\)
= 4
Therefore, there cannot be a perfect square number that is negative.
The square root of 4 should give us both the values 2 and -2 as results. However, the radical symbol used ” \(\sqrt{}\) “ conventionally considers only the positive square root and not the negative. Hence, \(\sqrt{4}\) = 2 and not – 2.