Square Root of 0.45

The square root is the inverse of the square of the number. 0.45 is not a perfect square. The square root of 0.45 is expressed in both radical and exponential form. In the radical form, it is expressed as √0.45, whereas (0.45)^(1/2) in the exponential form. √0.45 = 0.67082, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 0.45 can be expressed in terms of its prime factors.

Step 1: Express 0.45 as a fraction: 45/100.

Step 2: Find the prime factors of 45 and 100. 45 = 3 x 3 x 5 (3² x 5) 100 = 2 x 2 x 5 x 5 (2² x 5²)

Step 3: Simplify √(45/100) using the prime factors. √(45/100) = √(3² x 5) / √(2² x 5²) = (3√5)/(10)

Since 0.45 is not a perfect square, finding the square root using prime factorization in a simplified form is limited to this expression.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Pair the digits of 0.45 starting from the decimal point, making it 45 (the equivalent of 45/100).

Step 2: Find a number whose square is less than or equal to 45. Since 6 x 6 = 36 and 7 x 7 = 49, take 6.

Step 3: Subtract 36 from 45, giving a remainder of 9.

Step 4: Bring down 00 to make the new dividend 900.

Step 5: Double the divisor (6) to get 12, and find a digit ‘d’ such that 12d x d ≤ 900. The digit is 7 (127 x 7 = 889).

Step 6: Subtract 889 from 900 to get the remainder 11.

Step 7: Add a decimal point and bring down 00 to make it 1100, and repeat the process to get more decimal places.

So the square root of √0.45 is approximately 0.67082.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 0.45 using the approximation method.

Step 1: Identify the closest perfect squares between which 0.45 lies. The smallest perfect square less than 0.45 is 0.36 (0.6²), and the largest perfect square greater than 0.45 is 0.49 (0.7²).

Step 2: Use linear approximation: (0.45 - 0.36) / (0.49 - 0.36) = 0.09 / 0.13 ≈ 0.692 Using this, the estimated square root is approximately 0.6 + 0.692 x (0.1) ≈ 0.6692.

So the square root of 0.45 is approximately 0.67082.

• Square root: A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 0.45, 7.86, 8.65, and 9.42 are decimals.
   
• Fraction: A fraction represents a part of a whole or, more generally, any number of equal parts. It is represented as p/q, where p and q are integers, and q ≠ 0.
   
• Approximation: Approximation is the process of finding a value that is close enough to the right answer, usually with some thought or calculation.