Square Root of 6.5

The square root is the inverse of the square of the number. 6.5 is not a perfect square. The square root of 6.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √6.5, whereas (6.5)^(1/2) in the exponential form. √6.5 ≈ 2.54951, 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 6.5 is broken down into its prime factors, if applicable.

Step 1: Finding the prime factors of 6.5 Since 6.5 is a decimal number, it can't be directly factorized like whole numbers. However, if considered as 65/10, it can be factorized as 5 x 13 / 2 x 5.

Step 2: As 6.5 is not a perfect square, calculating 6.5 using prime factorization for the square root is not directly feasible.

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 consider 6.5 as 6500 (by multiplying by 100 to avoid decimals), and group the numbers from right to left as 65 and 00.

Step 2: Find n whose square is less than or equal to 65. We can say n is 8 because 8 x 8 = 64, which is less than 65. Now the quotient is 8, and the remainder is 65 - 64 = 1.

Step 3: Bring down the next pair of zeros making the new dividend 100.

Step 4: Add the old divisor with the same number: 8 + 8 = 16, which will be our new divisor.

Step 5: Find 2n × n ≤ 100. Let n be 0, then 160 x 0 = 0.

Step 6: Subtract 0 from 100, the difference is 100, and the quotient becomes 8.0.

Step 7: Add a decimal point and two zeros to the dividend, making it 10000.

Step 8: Find the new divisor, which is 1600. Assuming n is 6, 1606 x 6 = 9636.

Step 9: Subtract 9636 from 10000, the result is 364.

Step 10: Continue doing these steps until you get the desired precision.

The square root of √6.5 is approximately 2.5495.

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 6.5 using the approximation method.

Step 1: Find the closest perfect squares around 6.5.

The smallest perfect square less than 6.5 is 4, and the largest perfect square greater than 6.5 is 9. √6.5 falls somewhere between 2 and 3.

Step 2: Use the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (6.5 - 4) / (9 - 4) = 2.5 / 5 = 0.5

Using this approximation, we identify the decimal point for our square root. Adding this to the smaller integer, we get approximately 2 + 0.5 = 2.5, but further refinement using the long division gives approximately 2.5495.

• Square root: A square root is the inverse of a square. Example: 2.5² = 6.25, and the inverse of the square is the square root, that is √6.25 ≈ 2.5.

• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Principal square root: A number has both positive and negative square roots; however, the positive square root, known as the principal square root, is often used in real-world applications.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 6.5, 7.2, and 9.42 are decimals.

• Long division method: A method used to find the square root of a non-perfect square by performing division steps until reaching the desired decimal precision.