What is the Square Root 4?
Finding the square and square root of a number are inverse operations. To find the square of a number you need to multiply the number by itself.
For example: To find the square of a number such as 2, we will multiply the number by itself, that is,
\(2\times2=2^2=4\)
A square root is represented by this radical symbol:‘√’.
To find the square root of 4 or \(\sqrt4,\)
Since, \(2^2=4\)
Therefore, \(\sqrt4=\sqrt{2^2}=2\)
Deriving Square Root using Prime Factorization
Steps to calculate the square root of 4 by prime factorization:
Step 1: Find the prime factorization of 4
\(\therefore\ 4=2\times2\)
Step 2: Make pairs of identical numbers.
\(4=\left(2\times2\right)\)
Step 3: Take a number from each pair and multiply.
Here, we only have 1 identical pair of 2. Hence,
\(\Rightarrow\sqrt4=\ 2\)
So, the square root of 4 is 2.
Each positive perfect square has two square roots, one is positive and the other is negative as their absolute value is the same.
Let us understand this with an example,
\(2\ \ \times\ \ 2=4\)
\(-2\ \times-2=4\)
From the above the square of -2 is 4 and the square of 2 is also 4, so we can say that square roots of 4 are 2 and -2.
\(\Rightarrow\ \sqrt4=\ \pm\ 2\)
In general,
Solved Square Root of Examples
Example 1: Max collected 20 identical square sheets of paper and the area of each sheet is \(\sqrt4\) m\(^2\). The square sheets are arranged as shown below. If he formed a rectangle out of those square sheets with width = 4 m, then, what would be the length of the rectangle?
Solution:
Area of one square sheets \(=\sqrt4=2\) m\(^2\)
Now, Max collected 20 such square sheets and created a rectangle out of it and the width of the rectangle is 4m. To find the length of the rectangle we will equate the area of the rectangle to the area of 20 square sheets.
Area of rectangle = Area of 20 square sheets
\(\Rightarrow\) Length x width = Number of square sheets x Area of one sheet
\(\Rightarrow\)Length x 4 = 20 x 2
\(\Rightarrow\) Length = \(\frac{40}{4}\)
\(\therefore\) Length = 10 m
Therefore, the length of the rectangle formed is 10 m.
Example 2: Evaluate \(5\sqrt4+2\sqrt4-20\)
Solution:
Given expression: \(5\sqrt4+2\sqrt4-20\)
We know that \(\sqrt4=2, \)
Hence, \(5\sqrt4+2\sqrt4-0=5\times2+2\times2-20\)
= 10 + 4 – 20
= -6
Hence the value of given expression \(5\sqrt4+2\sqrt4-20\) is -6.
Example 3: Find the value of square root of 400.
Solution:
\(\sqrt{400}=\sqrt{4\ \times\ 100}\)
\(=\sqrt{\left(2 \times 2\right)\times\left(2\times2\right)\times\left(5\times5\right)}\)
\(=2\times2\times5\)
= 20
Therefore, the square root of 400 is 20.
There are two square roots of 4 which are 2 and -2.
The value of the power when the exponent is \(\frac{1}{2}\) is the square root of the base. So in exponential form, the square root of 4, \(\sqrt4,\) is written as \(4^{\frac{1}{2}}.\)
When integers are the square root of the given number, then the given number is a perfect square. Therefore, 4 is a perfect square as \(\sqrt4=2\) and 2 is an integer.