Square Root of 6.4

The square root is the inverse of the square of a number. 6.4 is not a perfect square. The square root of 6.4 is expressed in both radical and exponential form. In the radical form, it is expressed as √6.4, whereas (6.4)^(1/2) in the exponential form. √6.4 ≈ 2.52982, 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, methods like the long division method and approximation method are used. Let us learn about 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 learn how to find the square root using the long division method, step by step.

Step 1: To begin, we need to group the numbers from right to left. For 6.4, treat it as 64 by considering two decimal places.

Step 2: Find n whose square is less than or equal to 6. We can say n = 2 because 2² = 4, which is less than 6. The quotient is 2, and after subtracting, the remainder is 2.

Step 3: Bring down 40 (from 4.0). Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: Find n such that 4n × n ≤ 240. Let n = 5, then 45 × 5 = 225.

Step 5: Subtract 225 from 240, and the remainder is 15.

Step 6: Since the remainder is less than the divisor, add a decimal point and two zeroes to the remainder to make it 1500.

Step 7: Find the new divisor that is 49 because 495 × 5 = 2475.

Step 8: Subtract 2475 from 1500 to get a new remainder. Continue this process until you reach the desired accuracy.

So the square root of √6.4 ≈ 2.53.

The approximation method is an easy method for finding the square roots of non-perfect squares. Here’s how to find the square root of 6.4 using approximation.

Step 1: Identify the closest perfect squares around 6.4.

The smallest perfect square below 6.4 is 4, and the largest perfect square above 6.4 is 9. √6.4 falls between 2 and 3.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). For 6.4, (6.4 - 4) / (9 - 4) = 0.48.

Add this decimal to the smaller perfect square root: 2 + 0.53 = 2.53.

Therefore, the square root of 6.4 is approximately 2.53.

• 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.

• Principal square root: A number has both positive and negative square roots, but the positive square root is more commonly used due to its applications in the real world. This is known as the principal square root.

• Decimal: A decimal is a number that has a whole number and a fraction separated by a decimal point, such as 7.86, 8.65, and 9.42.

• Long division method: A method used to find the square root of a number by dividing it into groups and using iterative steps to approximate the square root value.