What is Square Root 6?
Square root of number 6 is a number that when multiplied by itself equals 6. So, a square root of number 6 is a solution of the equation x\(^2\) = 6.
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 p, we will multiply the number by itself, that is,
p x p = p\(^2\)
Similarly, to find the square root of 6, we will do the inverse operation.
A square root is represented by the radical symbol:‘√’.
So, the square root of 6 is expressed as \(\sqrt{6}\).
How to Find the Square Root of 6?
The number 6 is not a perfect square. Therefore, \(\sqrt{6}\) will result in an irrational number.
So, the value of \(\sqrt{6}\) can be approximated to the nearest integer or to the nearest tenths, hundredths and so on.
Let’s find the square root of 6 using the prime factorization method.
Step 1: Find the prime factors of 6
∴ 6 = 2 × 3
Step 2: Make pairs of identical numbers.
Since we do not have any identical numbers in the prime factorization we will need to write 2 as \(\sqrt{2}\) x \(\sqrt{2}\) and 3 as \(\sqrt{3}\) x \(\sqrt{3}\)
Therefore,
6 = (\(\sqrt{2}\) x \(\sqrt{2}\) x \(\sqrt{3}\) x \(\sqrt{3}\))
Step 3: Take a number from each pair and multiply them to get square root.
\(\sqrt{6}\) = \(\sqrt{2}\) x \(\sqrt{3}\)
\(\Rightarrow\) \(\sqrt{6}\) = 1.414 x 1.732 Substitute \(\sqrt{2}\) \(\approx\) 1.414 and \(\sqrt{3}\) \(\approx\) 1.732
\(\sqrt{6}\) \(\approx\) 2.449
Therefore, the square root of 6 is approximately 2.449
What is the Negative Square Root of 6?
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,
2.449 × 2.449 \(\approx\) 6
-2.449 × -2.449 \(\approx\) 6
[Since, multiplying two negatives results in a positive)]
From the above the square of -2.449 is 6 and square of 2.449 is also 6, so we can write-
\(\Rightarrow\) \(\sqrt{6}\) = ± 2.449
Rapid Recall
Solved Square Root of 6 Examples
Example 1: Evaluate the expression (3\(\sqrt{6}\) x 2\(\sqrt{6}\)) -10
Solution:
Given expression: (3\(\sqrt{6}\) x 2\(\sqrt{6}\)) -10
3\(\sqrt{6}\) × 2\(\sqrt{6}\) = (6 × 6) -10 [Simplify]
= 36 – 10 [Multiply]
= 26 [Subtract]
Hence the value of given expression (3\(\sqrt{6}\) x 2\(\sqrt{6}\)) -10 is 26.
Example 2: The area of a crop circle is 188,400 square feet. What is the radius of the crop circle?
Solution:
To find the radius of the crop circle use the formula for the area of a circle.
The details provided:
Area of crop circle = 188,400 square feet
Area of a circle = \(\pi\)r\(^2\)
188400 = 3.14r\(^2\) [Substitute area as 188400 and \(\pi\) = 3.14]
60000 = r\(^2\) [Divide each side by 3.14]
\(\sqrt{60000}\) = r [Take the positive square root of each side]
\(\sqrt{6~\times~10000}\) = r [Rewrite 60000 as 6 x 10000]
\(\sqrt{6}\) x \(\sqrt{10000}\) = r
2.449\(\times\)100 = r [Substitute \(\sqrt{6}\) = 2.449 and \(\sqrt{10000}\) = 100]
244.9 = r [Multiply]
Hence, the radius of the crop circle is about 244.9 feet.
Example 3: Find the value of square root of 600.
Solution:
\(\sqrt{600}\) = \(\sqrt{6~\times~100}\) [Rewrite 600 as 6 x 100]
= \(\sqrt{6~\times~(2~\times~2)~\times~(5~\times~5)}\) [Prime factorize 100]
= \(\sqrt{6}\) x 2 x 5 [Simplify]
= 2.449 x 10 = 24.49 [Substitute \(\sqrt{6}\) as 2.449 and multiply]
Hence, the value of the square root of 600 is about 24.49.
[Note: 24.49 is an approximate value of square root of 600 as 2.449 is an approximate value of square root of 6.]
6 has two square roots just like any other number.
A number which cannot be expressed as a simple fraction is known as an irrational number.
No, 6 is not a perfect square number as the value of the square root of 6 is not an integer.
Prime factorization and long division methods are used to find out the square root of a number.
In the exponential form square root of 6 is represented as six raised to the power half or one by two.