Value of root 2

Well, the answer to this simple question is, the value of root 2 (i.e, √2)  is 1.412. To understand what is root 2 (√2)  and the calculation of the value of root 2, we first need to understand the concept of power and roots.

Powers and Roots  

Power: Let a be any number and n be a natural number,then 

{an}= a×a×a.........multiplied n times

Where a is called the base, and n is called the exponent or index, and an is called the exponential expression. an  is read as ‘a raised to the power n’ or ‘ a to the power n’ or simply ‘a power n.’

In particular, a¹ = a.

Root: When a number is multiplied by itself, the product so obtained is called the square of that number. Thus, the number itself becomes the root of its multiple (resultant).

For Example:

(1) \[2 \times 2 = 4\] i.e. \[{2^2} = 4\], so 2 is the root of 4.

(2) \[\frac{3}{5} \times \frac{3}{5} = \frac{9}{5}\] i.e. \[{\left( {\frac{3}{5}} \right)^2} = \frac{9}{5}\]is the root of\[\frac{9}{5}\].

If a is any number then a2 is called a square number or a perfect square number.

Square Root

The square root of a number is that number which when multiplied by itself gives the original number.

For Example:

1. As\[2 \times 2\]=4, square root of 4 is 2.

2. As \[\frac{3}{5} \times \frac{3}{5} = \frac{9}{{25}}\], square root of \[\frac{9}{{25}}\]is \[\frac{3}{5}\]

Method to find the square root of a given square number

Step 1) Express the given number as the product of primes.

Step 2)Make groups in pairs of the same prime.

Step 3) Take one factor from each pair of primes. Multiply them together; the product so obtained is the required square root of the given number.

Value of Root Two (\[\sqrt 2 \])

Let us take a number ‘a’ which when multiplied to itself gives ‘2’ as a result. So we can say that a is the square root of 2. That is, \[\sqrt 2  = a\]. 

To find the value of a, we first check if the given number is a perfect square or not. You can also refer to the table given above to find the square root of a perfect square. But since, 2 is not a perfect square and is also an irrational number, the exact value of under root 2 can never be determined. We can still find the approximate under root value 2 by the long division method.

Value of root two by ‘Long Division Method

Step 1)Choose a number by finding at least two roots that are perfect.

Step 2) Divide 2 by the chosen square roots

Step 3) Take the average of the result of the division and 2. 

Step 4) Use the result and repeat steps two and three until you get a satisfactory result.

\[\textrm{The value of}\sqrt{2}=1.41421356237309504880168872420969807856967187537694\]

The division goes on, but we take an approximate value of 2 that is 1.414.

Solved Examples

Example 1: Find the value of the following:

1) \[{7^4}\]

2) \[{\left( { - 3} \right)^5}\]

3) \[{\left( { - \frac{4}{5}} \right)^5}\]

4) \[{\left( { - 2\frac{2}{3}} \right)^2}\]

Solution:

1) \[{7^4} = 7 \times 7 \times 7 \times 7 = 2401\]

2) \[{\left( { - 3} \right)^5} = \left( { - 3} \right)\left( { - 3} \right)\left( { - 3} \right)\left( { - 3} \right)\left( { - 3} \right) =  - 243\]

3) \[{\left( { - \frac{4}{5}} \right)^5} = \left( { - \frac{4}{5}} \right)\left( { - \frac{4}{5}} \right)\left( { - \frac{4}{5}} \right)\left( { - \frac{4}{5}} \right)\left( { - \frac{4}{5}} \right) =  - \frac{{1024}}{{3125}}\]

4) \[{\left( { - 2\frac{2}{3}} \right)^2} = {\left( { - \frac{8}{3}} \right)^2} = \left( { - \frac{8}{3}} \right)\left( { - \frac{8}{3}} \right) = \frac{{64}}{9} = 7\frac{1}{3}\]

Example 2: Find the square root of the following numbers:

1) 6\[\frac{1}{4}\]

2) 42.25

Solution:

1) \[\sqrt {6 \frac{1}{4}}\] = \[\sqrt {\frac{{25}}{4}}\]  = \[\frac{{\sqrt {5 \times 5} }}{{\sqrt {2 \times 2} }} = \frac{5}{2} = 2\frac{1}{2}\]

2) \[\sqrt {42.25}  = \sqrt {\frac{{4225}}{{100}}} \]

                 \[ = \sqrt {\frac{{169}}{4}} \]

          \[ = \frac{{\sqrt {13 \times 13} }}{{\sqrt {2 \times 2} }} = \frac{{13}}{2} = \frac{{13 \times 5}}{{2 \times 5}} = \frac{{65}}{{10}} = 6.5\]

Learn about Value of Root 2

This article covers the value of root 2 and also different methods to find out the square root of 2. The frequently asked questions at the end of this article can help you with your doubts if you encounter any while going through the same. An inverse operation of a square is a square root. In radical form Square root of 2 is represented as √2 and (2)1/2  in the exponential form. When √2 is rounded to 10 decimal points it gives 1.4142135624.

The Square Root of 2 in:-

Exponential form- (2)1/2

Radical form- √2

Square root- 1.4142135623730951