Square Root of 2.45

The square root is the inverse of the square of a number. 2.45 is not a perfect square. The square root of 2.45 is expressed in both radical and exponential form. In radical form, it is expressed as √2.45, whereas (2.45)^(1/2) is the exponential form. √2.45 ≈ 1.565, 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, for non-perfect square numbers, the long division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation method

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: To begin with, we need to pair the numbers from right to left. For 2.45, we consider 2.45 as a whole.

Step 2: Find a number whose square is closest to 2.45. In this case, the number is 1 because 1 × 1 = 1, which is less than or equal to 2.45. The quotient is 1.

Step 3: Subtract 1 from 2.45, giving a remainder of 1.45. Bring down the next pair of zeros to make it 145.

Step 4: Double the divisor (1) to get 2, which forms part of the new divisor. Now, find a digit n such that 2n × n is less than or equal to 145. Here, n is 5, since 25 × 5 = 125.

Step 5: Subtract 125 from 145, resulting in 20. Bring down another pair of zeros, making it 2000.

Step 6: The new divisor is 30 (from 25) plus a digit n. Finding n, we get 6 since 306 × 6 = 1836.

Step 7: Subtract 1836 from 2000, getting a remainder of 164. The quotient is 1.56.

Step 8: Continue these steps until the desired accuracy is achieved.

The square root of 2.45 is approximately 1.565.

The approximation method is another approach to find 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 2.45 using the approximation method.

Step 1: Identify the closest perfect squares around 2.45. The closest perfect square less than 2.45 is 1 (√1 = 1) and the closest perfect square greater than 2.45 is 4 (√4 = 2).

Step 2: Use interpolation: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Applying the formula: (2.45 - 1) / (4 - 1) = 1.45 / 3 ≈ 0.4833.

Step 3: Add this value to the square root of the smaller perfect square: 1 + 0.4833 ≈ 1.4833. This approximation shows that the square root of 2.45 is approximately 1.565, which can be refined further.

• 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: A decimal is a number that has a whole number and a fractional part separated by a decimal point. Examples: 7.86, 8.65, 9.42.
   
• Long division method: A technique used to find the square root of a non-perfect square by dividing the number into pairs of digits and solving step-by-step.
   
• Approximation method: A method used to estimate the value of a square root by comparing it to nearby perfect squares and interpolating between them.