Square root of 75

The square root of 75 is ±8.66025403784. As defined, the square root is just the inverse of squaring a number, so, squaring 8.66025403784 will result in 75.  The square root of 75 is expressed as √75 in radical form. In exponential form, it is written as (75)^1/2  .
 

We can find the square root of 75 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method
   

The prime factorization of 75 is done by dividing 75 by prime numbers and continuing to divide the quotients until they can’t be separated anymore.

Steps for Prime Factorization of 75:

Step 1:Find the prime factors of 75.

Step 2: After factorizing 75, make pairs out of the factors to get the square root.

Step 3: 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 75 = 5 × 5 ×3  

For 75, only one pair of factors can be obtained, but a single 3 is remaining.

So, it can be expressed as  √75 =   √(5 × 5×3) = 3√5

3√5 is the simplest radical form of √75

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 75:

Step 1 : Write the number 75, 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 75. Here, it is 8, Because 8²=64 < 75.

Step 3 : Now divide 75 by 8 (the number we got from Step 2) such that we get 8 as quotient and then multiply the divisor with the quotient, we get 64. Subtract 64 from 75, we get 11. Add a decimal point after the quotient 8, and bring down two zeroes and place it beside 11 to make it 1100.

Step 4: Add 8 to same divisor, 8. We get 16.

Step 5: Now choose a number such that when placed at the end of 16, a 3-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 1100. Here, that number is 6. 

166×6=996<1100.

Step 6: Do 1100-996=104. Again, bring down two zeroes and make 104 as 10400. Simultaneously add the unit’s place digit of 166, i.e., 6 with 166. We get here, 172. 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, 4400 (refer to the picture), after some iterations and keeping the division till here, at this point 
             
Step 7 : The quotient obtained is the square root. In this case, it is 8.660….

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 75. Here, it is 64 and 81.

Step 2: We know that, √64=8 and √81=9. This implies that √75 lies between 8 and 9.

Step 3: Now we need to check √75 is closer to 8 or 9. Let us consider 8.5 and 9. Since (8.5)²=72.25 and (9)²=81. Thus, √75 lies between 8.5 and 9.

Step 4: Again considering precisely, find squares of (8.6)²=73.96 and (8.8)²= 77.44.

We can iterate the process and check between the squares of 8.6 and 8.7 and so on.

We observe that √75=8.660…

• 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, 2⁴ = 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