What is the square root of 75? How to find the square root of 75?

The square root of 75 is 8.66. The square root of a number is determined when a quantity of a number is produced when multiplied by itself or a factor of a number when multiplied by itself gives the original number. 

The square root is mentioned with a symbol √. The square root of  75 is denoted using √75  or 751/2. In the radical form, the square root of 75 is represented as  5√3.

In simple terms when a value is multiplied by itself it gives the original number. For a better understanding. The number 20 when multiplied by itself
(20×20) = 400. The square root for (√400) is 20.

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Different ways of writing Square Root 

Square root of 75 = √75 = 5√5
The square root of 75 in decimal form,
√75=  8.66
The square root of 75 in exponent form, 
75^1/2= 8.66

What is the Square root of 75?

The square root of a number when multiplied itself gives the original number. The square root of 75 is expressed as √75
√75 = √(Number)×√(Number)
√75 = (8.66×8.66)
√75 = √(8.66)2
Now, remove the square on the right-hand side we get.
√75 = 8.66
Thus, by multiplying 8.66 we get the number 75.

The square root of 75 in radical form

The number 75 is a perfect cube number and hence can be expressed in radical form.
√75 = √3√5√5
√75 = √3 √5²
√75 = 5√3

The calculation for the square root of 75

To find the square root of any number first the given number is usually checked to determine whether they are a perfect square number or not. The square root of the number is found using the long division method.

For the numbers with a perfect square, it is easy to find the square roots and it is a bit tough for non-square values. Numbers like 4,9, 16,25, etc., are perfect squares. Numbers such as  2,3,7,18,75 and 80 are not perfect square numbers.
75 is not a rational number or a perfect square number.

Prime factorization cannot be used to find the square root of 75. Hence, the long division method is used to find the square root of  75.

Methods to find the square root

There are three methods to find the square root of a number

• Prime factorization method
• Long division method
• Repeated subtraction

These methods described above do not apply to finding the square root of any number.  

Prime factorization method

To find the square root of a number using the prime factorization method first knows the prime factors of the numbers. Let us take n as a prime number, by grouping their squares we get n² now multiplying them we get the square root of the number. 

• Step 1: Find the prime factor for the given number
  √75 = 3×5×5
• Step 2: Pair the prime factors.
  √75 = 3×52
  Take square on both sides
  √75=√3×√52
  You can cancel the square with the square root we get
  √75 = 5√3
• Step 3: Multiply the factors
  The √3 = 1.732
  √75 = 5×1.732
  √75 = 8.66

Long division method

Long division is one of the easiest methods to find the square root of any number. It was the preferred method to be used for the non-perfect square numbers. Find the integer that can divide the number and proceed with the long division. 

Steps to find the square root of 75:

• Step 1: Find the smallest integer that can divide the number. 
• Step 2: Keep following the long division using divisor and dividend. 
• Step 3: When the particular number of satisfaction is reached the quotient is the square root of the number.  

Find the square root of 75

• Step 1: Find the smallest integer that can divide the number. 75 is not a perfect root number and 8 is the nearest number that can divide it. 
• Step 2: Keep following the long division using divisor and dividend. 

• Step 3: When the particular number of satisfaction is reached the quotient is the square root of the number.
• Hence, the √75 = 8.66

Solved Examples

Example 1: Find the square root of 13 using the repeated subtraction method.
Solution:
13 – 1 = 12
12 – 3 = 9
9-5=4
4-7=-3
The value in the last step should be zero.
Here, the end value is a negative zero.
Hence, the square root of 13 cannot be found using repeated subtraction methods.

Example 2: What is the square root of 75 using the long division method?
Solution:

The √75 = 8.66

Example 3: What is the square root of 75 using the prime factorization method?
Solution:

• Step 1: Find the prime factor for the given number
  75 = 3×5×5
• Step 2: Pair the prime factors.
  75 = 3×5²
  Take square on both sides
  √75=√3×5²
  You can cancel the square with the square root we get
  √75 = 5√3
• Step 3: Multiply the factors
  The √3 = 1.732
  √75 = 5×1.732
  √75 = 8.66

Example 4: What is an irrational number?
Solution:
A number is said to be an irrational number if it cannot be expressed in the form of a ratio or fractions. 
Example: √2, √3, √5 or √75

Example 5: Solve 2√75
Solution:
√75 = 8.66
2√75 = 2×8.66
2√75 = 17.32.

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FAQs on the square root of 75