What is the Square Root 12?
The square root of number 12 is a number which when multiplied by itself results in 12. So, the square root of number 12 is a solution of the equation \(x^2\) = 12.
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 4, we will multiply the number by itself, that is,
4 × 4 = 16
Whereas, to find the square root of 16, we will do the inverse operation.
A square root is represented by the radical symbol: ‘√’.
Since, \(4^2\) = 16
\(\Rightarrow \sqrt{16}=\sqrt{4^2}\) = 4
How to Find the Square Root of 12?
The number 12 is not a perfect square. Therefore, \(\sqrt{12}\) will result in an irrational number.
So, the value of \(\sqrt{12}\) can be approximated to the nearest integer or to the nearest tenth, hundredth and so on.
Let’s find the square root of 12 using the prime factorization method.
Step 1: Find the prime factors of 12
∴ 12 = 2 × 2 × 3
Step 2: Make pairs of identical numbers.
Since we do not have a pair of 3 we can write 3 as \(\sqrt{3}\) x \(\sqrt{3}\)
12 = \((2\times2)\times(\sqrt{3}\times\sqrt{3})\)
Step 3: Take a number from each pair and multiply them to get square root.
\(\sqrt{12}\) = 2 × \(\sqrt{3}\)
\(\Rightarrow \sqrt{12}\) = \(2\sqrt{3}\)
Since the value of \(\sqrt{3}\) = 1.732 (approx.)
\(\Rightarrow \sqrt{12}\) = 2 × 1.732 = 3.464 (approx.)
So, the square root of 12 is \(2\sqrt{3}\) or about 3.464
Definition of Negative Square Root
Each positive number has two square roots, one is positive and other is negative both having the same absolute value.
Let us understand this with example,
\(2\sqrt{3}\times2\sqrt{3}=12\)
\((-2\sqrt{3})\times(-2\sqrt{3})=12\)
[Since, multiplying two negatives results in a positive)]
From the above the square of \(-2\sqrt{3}\) is 12 and square of \(2\sqrt{3}\) is also 12, so we can say that square roots of 12 are \(2\sqrt{3}\) and \(-2\sqrt{3}\) both.
\(\sqrt{12}\) = ± \(2\sqrt{3}\)
In general,
Rapid Recall
Solved Square Root of 12 Examples
Example 1: The area of a square crop field is 1200 \(m^2\). Find the side length and perimeter of the field.
Solution:
Let the side of the square crop field be ‘a’ m.
The area of the square field = \(a^2\) [Formula for area of a square]
\(\Rightarrow\) 1200 = \(a^2\) [Given area = 1200\(m^2\)]
\(\Rightarrow\sqrt{1200}\) = \(\sqrt{a^2}\) [Taking square root on both sides]
\(\Rightarrow\sqrt{12\times100}\) = \(\sqrt{a^2}\) [Rewrite 1200 as 12 x 100]
\(\Rightarrow\sqrt{12\times(2\times2)\times(5\times5)}\) = a [Prime factorize 100]
\(\Rightarrow\sqrt{12}\times\sqrt{(2\times2)\times(5\times5)}\) = a
\(\Rightarrow\sqrt{12}\times2\times5\) = a [Simplify]
\(\Rightarrow3.464\times10\) = a [Substitute 3.464 for \(\sqrt{12}\)]
∴ 34.64 = a [Simplify]
Hence, the side of the square crop field is about 34.64 m.
Now,
Perimeter of a square field = 4 × a [Formula for perimeter of a square]
= 4 × 34.64 [Substitute a = 34.64]
= 138.56 m
Therefore, the length of the side of the crop field is 34.64 m and its perimeter is about 138.56 m.
Example 2: Find the value of 5 multiplied by \(\sqrt{12}\).
Solution:
5 × \(\sqrt{12}\)
Substitute 3.464 for \(\sqrt{12}\)
5 × \(\sqrt{12}\) = 5 × 3.464
= 17.32 [Multiply]
Hence the value of 5 multiplied by \(\sqrt{12}\) is about 17.32.
Example 3: Find the value of \((3\sqrt{12}\times\sqrt{3})+(\sqrt{12}\times\sqrt{3})\) .
Solution:
Given expression: \((3\sqrt{12}\times\sqrt{3})+(\sqrt{12}\times\sqrt{3})\)
\((3\sqrt{12}\times\sqrt{3})+(\sqrt{12}\times\sqrt{3})\) = \((3\times2\sqrt{3}\times\sqrt{3})+(2\sqrt{3}\times\sqrt{3})\) [Substitute \(\sqrt{12}\) = \(2\sqrt{3}\)]
= (6 × 3) + (2 × 3) [Simplify]
= 18 + 6 [Multiply]
= 24 [Add]
Therefore, the value of the given expression is 24.
12 has two square roots just like any other number.
No, 12 is not a perfect square number.
The square root of 12 is = 3.464101615137………
So, the square root 12 is an irrational number because it is a non-terminating and non-repeating value.