Square Root of 13

The square root of 13 is ±3.60555127546. Finding the square root is just the inverse of squaring a number and hence, squaring 3.60555127546 will result in 13.  The square root of 13 is written as √13 in radical form.
 

We can find the square root of 13 through various methods. They are:

 

• Prime factorization method

 

• Long division method

 

• Approximation/Estimation method

The prime factorization of 13 is done by dividing 13 by prime numbers and continuing to divide the quotients until they can’t be separated anymore.

 Find the prime factors of 13

 

 

After factorizing 13, make pairs out of the factors to get the square root.

 

 If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

 

So, Prime factorization of 13 = 13 × 1  

 

But here in case of 13, no pair of factors can be obtained but a single 13 is remaining

 

So, it can be expressed as  √13 =   √(13 × 1) = √13

 

√13 is the simplest radical form of √13

 

This is a method used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

 

Follow the steps to calculate the square root of 13:

Step 1 : Write the number 13, and draw a horizontal bar above the pair of digits from right to left.

Step 2 : Now, find the greatest number whose square is less than or equal to 13. Here, it is 3, Because 32=9 < 13.

Step 3 : Now divide 13 by 3, such that we get 3 as quotient and then multiply the divisor with the quotient, we get 9.

Step 4: Add a decimal point after the quotient 3, and bring down two zeroes and place it beside the difference 4 to make it 400.

Step 5: Add 3 to same divisor, 3. We get 6.

Step 6: Now choose a number such that when placed at the end of 6, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 400. Here, that number is 6. 

66×6=396<400.

Step 7: Subtract 400-396=4. Again, bring down two zeroes and make 4 as 400. Simultaneously add the unit’s place digit of 66, i.e., 6 with 66. We get here, 72. Apply Step 5 again and again until you reach 0. 

 

We will show two places of precision here, and so, we are left with the remainder, 3975 (refer to the picture), after some iterations and keeping the division till here, at this point 

             
Step 8 : The quotient obtained is the square root. In this case, it is 3.605….

 

Approximation or estimation of square root is not the exact square root, but it is an estimate.

Here, through this method, an approximate value of square root is found by guessing.

 

Follow the steps below:

Step 1: Find the nearest perfect square number to 13. Here, it is 9 and 16.

Step 2: We know that, √9=3 and √16=4. This implies that √13 lies between 3 and 4.

 

 

Step 3: Now we need to check √13 is closer to 3 or 4. Let us consider 3.5 and 4. Since (3.5)²=12.25 and (4)²=16. Thus, √12 lies between 3 and 3.5.

 

 

Step 4: Again considering precisely, we see that  √13 lies close to (3.5)2=12.25. Find squares of (3.6)²=12.96 and (3.8)²= 14.44.

 

 

We can iterate the process and check between the squares of 3.602 and 3.7 and so on.

We observe that √13=3.605…

 

• Exponential form:  An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 2 4 = 16, where 2 is the base, 4 is the exponent 

 

• Prime Factorization:  Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3

 

• Prime Numbers: Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

 

• Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

 

• Perfect and non-perfect square numbers: Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24