Square Root of 5.65

The square root is the inverse of the square of the number. 5.65 is not a perfect square. The square root of 5.65 is expressed in both radical and exponential form. In the radical form, it is expressed as √5.65, whereas (5.65)^(1/2) in the exponential form. √5.65 ≈ 2.377, 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 prime factorization method is typically used for perfect squares, and as 5.65 is not a perfect square, we cannot use this method effectively. Therefore, calculating 5.65 using prime factorization is impractical.

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: Set up 5.65 for division from right to left, grouping the digits. In this case, it remains as 5.65.

Step 2: Find n whose square is less than or equal to 5. The closest perfect square is 4, so n = 2. The quotient is 2, and the remainder is 5 - 4 = 1.

Step 3: Bring down 65 next to the remainder to form 165.

Step 4: Double the quotient (2) to get 4, and determine a digit x such that 4x * x ≤ 165. By testing, x = 3 works because 43 * 3 = 129, which is less than 165.

Step 5: Subtract 129 from 165 to get a remainder of 36.

Step 6: Add a decimal point and bring down 00, making it 3600.

Step 7: New divisor is 46x, find x such that 46x * x ≤ 3600. Through testing, x = 7 fits because 467 * 7 = 3269

. Step 8: Subtract 3269 from 3600 to get 331.

Step 9: Continue this process to get more decimal places if needed. For practical purposes, we stop here with a quotient of 2.377.

So, √5.65 ≈ 2.377.

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 5.65 using the approximation method.

Step 1: Determine the closest perfect squares.

The closest squares are 4 (2²) and 9 (3²). √5.65 falls between 2 and 3.

Step 2: Apply interpolation to approximate the value: (5.65 - 4) / (9 - 4) = 1.65 / 5 = 0.33

Using this approximation, √5.65 ≈ 2 + 0.33 = 2.33. However, refine using further calculations or more accurate methods to estimate √5.65 ≈ 2.377.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse square root is √16 = 4.

• Irrational number: An irrational number cannot be written as a simple fraction; its decimal goes on forever without repeating.

• Decimal: A decimal is a number that includes both a whole number and a fraction separated by a decimal point, such as 7.86.

• Perfect square: A perfect square is an integer that is the square of another integer. For example, 25 is a perfect square because it is 5².

• Long division method: A technique used to find the square root of numbers that are not perfect squares by performing division in a step-by-step manner.