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Value of root 2

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Last updated date: 29th Mar 2024
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What is the 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.


Number

Perfect Square

Number

Perfect Square

1

1²=1

11

11²=121

2

2²=4

12

12²=144

3

3²=9

13

13²=169

4

4²=16

14

14²=196

5

5²=25

15

15²=225

6

6²=36

16

16²=256

7

7²=49

17

17²=289

8

8²=64

18

18²=324

9

9²=81

19

19²=361

10

10²=100

20

20²=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 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

FAQs on Value of root 2

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.

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

3. What is meant by irrational numbers?

The real numbers that are not rational or cannot be represented in the form of a ratio are irrational numbers. Such as p/q in which p and q are integers,q≠0.  Irrational numbers are non-recurring and non-terminating decimals. In addition, subtraction division of multiplication of two irrational numbers cannot necessarily be a rational number. When a rational and an irrational number are added than their sum is always an irrational number only.

4. What is meant by rational numbers?

The word "rational" has originated from the word "ratio".  So if a number can be expressed in the fractional form in which the numerator and the denominator are integers then the number is a rational number. Any number can be a rational number if expressed as a fraction. Whole numbers, natural numbers, integers and terminating decimals are all rational numbers. These numbers can be added, subtracted, multiplied and divided like fractions.

5. What is factor and factorization?

Factors are those numbers that keep getting multiplied by one another. For example, the factors of 16 are 1, 2, 4, 8 and 16 because all these numbers are positive whole numbers that can divide evenly into 16. The prime factorisation method is used to find prime factors of a composite number. In short prime factorization of any number means to represent that particular number as the product of primes.

6. What is the division method in prime factorization?

Prime factorization can be done using the division method that has been explained as follows:


Division Method - In this method we try to divide the given number by the smallest prime number that divides it completely. And then again the quotient is divided by the smallest prime number and this repeats until the quotient becomes 1. All the devices are then multiplied top to get an answer.