What is the square root of 45? How to find the square root of 45?

The square root of 45 is 6.708. The square root is mentioned with a symbol √. The square root of  45 is denoted using √45  or 451/2. In the radical form, the square root of 45 is represented as 3√5.

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. 

In simple terms when a value is multiplied by itself it gives the original number. For a better understanding. The number 9 when multiplied by itself (9× 9) = 81. The square root for  (√81) is 9.

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Different ways of writing Square Root 

Square root of √45 = √45±3√5
The square root of 45 in decimal form,
√45=  6.708
The square root of 45 in exponent form, 
45^1/2= 6.708.

The calculation for the square root of 45

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, 80, and 45 are not perfect square numbers.
45 is not a rational number or a perfect square number.

Prime factorization cannot be used to find the square root of 45. Hence, the long division method is used to find the square root of 45.

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
  45 = 3×3×5
• Step 2: Pair the prime factors.
  45 = 3² × 5
  Take square on both sides
  √45=√5×√3²
  You can cancel the square with the square root we get
  √45 = 3√5
• Step 3: Multiply the factors
  The √5 = 2.236
  √45 = 3×2.236
  √45 = 6.708

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 45:

• 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 45

• Step 1: Find the smallest integer that can divide the number. 45 is not a perfect root number and 6 is the nearest number that can divide it. 
• Step 2: Keep following the long division using divisor and dividend.

Long division method for the square root of 45

Step 3: When the particular number of satisfaction is reached the quotient is the square root of the number.
Hence, the √45 = 6.708

Solved Examples

Example 1: Find the square root of 16 using the prime factorization method
Solution:
Prime factor of 16 = 2×2×2×2 or 2⁴
Now let us take the similar squares 

Now, multiply the squares 2×2 = 4
Hence, the square root of 16 is
√16 = 4.

Example 2: What is the square root of 45 using the long division method?
Solution:

The √45 = 6.708

Example 3: Solve 2a+45=9
Solution:
2a+6.708 = 9
2a = 9- 6.708
2a = 2.292
a = 2.292÷2
A = 1.146

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 √18

Example 5: Solve √4×√45
Solution:
The √45 = 6.708
√4 = 2
√4×√45 = 2× 6.708
√4×√45 = 13.416

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