Value of root 2

What is the value of root 2?

Well, the answer to this simple question is, the value of root 2 (i.e, \[\sqrt 2 \])  is 1.412. To understand what is root 2 ( \[\sqrt 2 \])  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

\[{a^n} = a \times a \times a..........\]multiplied n times

Where a is called the base, and n is called the exponent or index, and \[{a^n}\]is called the exponential expression. \[{a^n}\]is read as ‘a raised to the power n’ or ‘ a to the power n’ or simply ‘a power n.’

In particular, \[{a^1} = a\].


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.



Number

Perfect Square

Number

Perfect Square

1

\[{1^2} = 1\]

11

\[{11^2} = 121\]

2

\[{2^2} = 4\]

12

\[{12^2} = 144\]

3

\[{3^2} = 9\]

13

\[{13^2} = 169\]

4

\[{4^2} = 16\]

14

\[{14^2} = 192\]

5

\[{5^2} = 25\]

15

\[{15^2} = 225\]

6

\[{6^2} = 36\]

16

\[{16^2} = 256\]

7

\[{7^2} = 49\]

17

\[{17^2} = 289\]

8

\[{8^2} = 64\]

18

\[{18^2} = 324\]

9

\[{9^2} = 81\]

19

\[{19^2} = 361\]

10

\[{10^2} = 100\]

20

\[{20^2} = 400\]


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 long division method.

Value of root two by ‘Long Division Method:


Step 1)Choose a number by finding at least two roots which 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 step two and three until you get a satisfactory result.



\[The{\text{ }}value{\text{ }}of\sqrt 2 \; = {\text{ }}1.41421356237309504880168872420969807856967187537694 \ldots \]


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

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

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

FAQ (Frequently Asked Questions)

1. Is root two a polynomial?

A polynomial is an algebraic expression having many terms in the form of constant (like 4) and variables (like x or y) which are raised to a non-negative integral power. But root two has no variable. Thus, root two is not a polynomial.

Can root two be written as a fraction?

 Let us suppose root two is a fraction,  

In that case, we can represent it as x/y where both x and y are whole numbers.

The fraction x/y to be the root of two needs to be multiplied to itself to give number 2.

2 = {x/y} x {x/y}

= (x/y)2

= x2/y2


Thus, we can also say that y2x 2 = x2

According to the even-odd rule if x2 is even then m must also be even and vice versa.

Now, we know that all even numbers are actually a multiple of 2

Thus, m is also a multiple of 2 which makes x2 as a multiple of 4.