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The square root of a number a is the number b such that bÂ² = a. The square root of any number is represented by the symbol and is also often known as radical. The number or expression given under the square root symbol is known as radicand. For example, in the expression \[\sqrt{3x + 5}\], the radicand is 3x + 5.

The square root is a commonly used function in Mathematics. It is widely used in subjects like Mathematics and Physics. Sometimes it is tedious to find the square root of a number, especially the numbers which are not the square of a number.

In this article, we will discuss the square root of 6, and how to calculate the root 6 value using the simplifying square root method.

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The square root of a number 6 is a number y such that yÂ² = 6. The square root of 6 in radical form is written as \[\sqrt{6}\].

The square root of 6 in radical form is expressed as \[\sqrt{6}\].

We can calculate the under root 6 value using different methods of square roots. These methods can be the long division methods, prime factorization method, or simplifying square root method. Let us discuss how to calculate the under root 6 value using the simplifying square root method.

To simplify a square root, make the number under the square root as small as possible while keeping it as a whole number. Mathematically, it can be expressed as:

\[\sqrt{x.y} = \sqrt{x} \times \sqrt{y}\]

To express the square root of 6 in the simplest form, we will make the number 6 as small as possible, ensuring to keep it as a whole number. Hence, the square root of 6 in simplest form is represented as:

\[\sqrt{6} = \sqrt{2} \times \sqrt{3}\]

This can be further simplified by substituting the value of \[\sqrt{2}\] and \[\sqrt{3}\].

\[\sqrt{6} = \sqrt{2} \times \sqrt{3}\]

\[\sqrt{6} = 1.414 \times 1.732\]

\[\sqrt{6} = 2.449\]

Hence, the square root of 6 in simplest form is 2.449.Â Similarly, we can also calculate the square root of any other whole numbers and their factors. Hence, simplifying the square root method is the simplest method of calculating the square root. We can also calculate the value of under root 6 using the calculator as this will give us the exact value. The exact value of the square root will always be given in a decimal number as it is impossible to determine a positive whole integer as a root for non rational numbers.

Following are the steps to simplify the square root using the perfect square method:

Find the perfect square that divides the number in the radicand.

Express the numbers as a factor of a perfect square.

Simplify the radicals.

Example

Simplify \[\sqrt{300}\]

Solution:

\[\sqrt{300} = \sqrt{100 \times 3}\]

\[\sqrt{300} = \sqrt{10 \times 10 \times 3}\]

\[\sqrt{300} = 10 \sqrt{3}\]

Hence, \[\sqrt{300}\] can be simplified as \[10\sqrt{3}\]

Simplify the following radical expressions :

\[\sqrt{48}\]

\[\sqrt{75}\]

Solutions:

1. \[\sqrt{48}\]

Step 1: The perfect square 16 will divide the number 48.

Step 2: Express 48 as a factor of 16

\[48 = 16 \times 3\]

Step 3: Reduce the square root of 16 as shown below:

\[\sqrt{48} = \sqrt{16 \times 3}\]

\[\sqrt{48} = \sqrt{4 \times 4 \times 3}\]

\[\sqrt{48} = 4\sqrt{3}\]

Hence, \[\sqrt{48}\] can be simplified as \[4\sqrt{3}\]

2. \[\sqrt{75}\]

Step 1: The perfect square 25 will divide the number 75.

Step 2: Express 75 as a factor of 25.

\[75 = 25 \times 3\]

Step 3: Simplify the radicals as shown below:

\[\sqrt{75} = \sqrt{25 \times 3}\]

\[\sqrt{75} = \sqrt{5 \times 5 \times 3}\]

\[\sqrt{75} = 5\sqrt{3}\]

Hence, \[\sqrt{75}\] can be simplified as \[5\sqrt{3}\].

FAQ (Frequently Asked Questions)

Q1. What is the Square Root of 6 in the Simplest Radical Form?

Ans: The square root of any number can only be simplified if the given number is divisible by a perfect square (other than number 1).

âˆš12 can be simplified further because 12 can be divisible by a perfect square i.e. 4.

Similarly, âˆš250 can also be simplified further because 250 can be easily divided by perfect square 25.

But, âˆš6 cannot be simplified further as 6 cannot be divided by a perfect square.

Hence, the square root of 6 in the simplest radical form is expressed as âˆš6.

Q2. Is the Value of Under Root 6 Rational or Irrational?

Ans: The value of under root 6 is considered as an irrational number because the number 6 is not a perfect square. This implies that the square root of 6 will have an infinite number when expressed in decimals. So, if we calculate the value of 6, we get the numbers that will go on into infinity, and will never repeat or terminate. It cannot be written even in the form of x/y, where y is not equal to 0. Hence, the value of 6Â that we received is a non-terminating.

Q3. How to Express the Value Under the Square Root 6 in a Fraction Form?

Ans: As we know, the value under the square root 6 is an irrational number. Therefore, the root 6 value cannot be expressed in an exact fraction form. However, we can make the value under the square root 6 into an approximate fraction by rounding it to the nearest hundredth. The square root of 6 to the nearest thousand is 2.45. Accordingly,

âˆš6 = 2.45/1

= 2.45/100

= 2 (9/20)

Hence, the approximate value of square root 6 in the fraction is 2 (9/20).