The Square Root Of 169

How to find Square Root of 169

Square of 13:

In mathematics, finding of square of any number is mostly like shooting fish in a barrel because when nosotros multiply the aforementioned number with itself and so we will get the square of that number.

For example:

Permit us suppose we accept to find the square of whatever number say Ten, and so we multiply X by itself i.e. 10 and we will go its foursquare as Y. Information technology can be written as (X)² = X*X= Y
In like way nosotros find the square of 13.
To find the square of xiii, we multiply 13 by the number itself i.e. by13 and nosotros write information technology as follows.
(13)² = 13*thirteen = 169

Foursquare root of 169:

Now, in contrary manner if we have to observe the square root of Y. The square root of Y is that unmarried value which when multiplied with itself gives the value Y.
That means, √Y = √(X*X) = X

Where √ is the symbol named as radical.

For instance:

The foursquare root of 169 tin can be written as,

√169 = √ (xiii*13) = xiii

Where √ is the symbol which is called as radical sign.

In short, we remember square of 13 and square root of 169 as

Annotation:

Every positive real number has two roots.
The square of any negative number is always the positive number.

For instance:

169 is the positive perfect foursquare which has two roots +13 and -thirteen also.
Only, the positive square root value is taken more often than not which is called equally principal square root or non-negative square root.
Hence, √169 = √(-13)*(-13) = -xiiiand √169 = √(xiii)*(xiii) = 13

Similarly,

(-13)*(-thirteen) = (-13)²= +169 and (+13)*(+13) = (+13)² = 169

Methods to find square root of perfect square like 169:

There are many methods to find the square root of perfect squares out of which we see the following method in detail.

Repeated Subtraction Method
Prime number factorization method

Repeated Subtraction Method:

In repeated subtraction method, we have to subtract the consecutive odd numbers starting from ane, from the perfect square number whose square root nosotros have to observe.
e. to find square root of 169, first we subtract 1 from it. 169 – ane = 168

And then adjacent odd number is 3, and then nosotros accept to subtract information technology from 168. 168 – 3 = 165

In this manner, we subtract the consecutive odd numbers from the corresponding values obtained after subtraction continuously till we become final value every bit 0.
And the value of number of odd numbers required to get 0 is the required square root.

For example:

We find the foursquare root of 169 by repeated subtraction method as follows:

169 – i = 168

168 – three = 165

165 – 5 = 160

160 – 7 = 153

153 – nine = 144

144 – 11 = 133

133 – thirteen = 120

120– 15 = 105

105 – 17 = 88

88– 19 = 69

69 – 21 = 48

48 – 23 = 25

25 – 25 = 0

Thus, here the total odd numbers used are 1, three, 5, 7, 9, 13, 13, fifteen, 17, xix, 21, 23, 25 and which are thirteen in numbers.
Hence, the foursquare root of 169 past repeated subtraction method is xiii.

Prime Factorization Method:

In prime factorization method, nosotros take to divide the perfect square number whose foursquare root we accept to find past prime starting from 2, 3, 5… then on till we become the remainder as 1.
Initially nosotros have to divide past prime number 2, if that number is not divisible past ii then we accept to have next prime i.due east. 3 and the process will be continued till we go remainder as 1.
Finally, we have to make pairs of the prime numbers taken in the class of multiplication and then we take to take its square root.

For Example:

Following is the process to find the square root of 169 by prime number factorization method.
As 169 is odd number hence it must be divisible past prime number two.
So, nosotros kickoff from prime number number 2 here.

169 ÷ thirteen = 13

13÷ xiii = 1

Thus, the prime number 13 used to go remainder as i are 13.

Thus, 169 = 13*13

And 169 = 13*13

By taking square root on both sides, we get

√169 = √(13*xiii) = √(xiii*13)

√169 = thirteen

Thus, we institute the square root of 169 as 13 by using prime number factorization method.

Multiple choice questions:

ane) The methods used to find foursquare root of perfect square 169 are

a) Prime factorization method

b) Repeated subtraction method

c) Both a and b

d) none

Ans: c) both a and b

two) In prime factorization method, the numbers used to divide the perfect square whose square root we have to discover are

a) Odd numbers

b) Even numbers

c) Successive prime numbers

d) Successive composite numbers

Ans: c) successive prime number numbers

3) The first prime number used to divide 169 is

a) 3

b) five

c) 11

d) thirteen

Ans: d) 13