Square Root of 65

The square root is the inverse of the square of the number. 65 is not a perfect square. The square root of 65 is expressed in both radical and exponential form.

In the radical form, it is expressed as √65, whereas (65)^1/2 in exponential form. √65 ≈ 8.06226, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 65 is broken down into its prime factors:

Step 1: Finding the prime factors of 65 Breaking it down, we get 5 x 13: 5¹ x 13¹

Step 2: Now we found out the prime factors of 65. The second step is to make pairs of those prime factors. Since 65 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 65 using prime factorization alone doesn't provide an exact integer result.

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 group the numbers from right to left. In the case of 65, we need to group it as 65.

Step 2: Now we need to find n whose square is less than or equal to 65. We can say n as ‘8’ because 8 x 8 = 64, which is less than 65. Now the quotient is 8, and after subtracting 64 from 65, the remainder is 1.

Step 3: Since the remainder is less than the divisor and we need more precision, we add a decimal point and bring down two zeros to make the new dividend 100.

Step 4: Double the quotient (8), making it 16, and find a number x such that 16x x ≤ 100. We find x as 0, so 160 x 0 = 0.

Step 5: Subtracting 0 from 100 gives us a remainder of 100.

Step 6: Bring down another set of zeros, making the new dividend 10000. Now, find x such that 1600x x ≤ 10000. We find x as 6, because 1606 x 6 = 9636.

Step 7: Subtract 9636 from 10000, leaving a remainder of 364.

Step 8: Continue this process to achieve the desired precision. The quotient builds up to approximately 8.06.

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 65 using the approximation method.

Step 1: We need to find the closest perfect squares to √65. The smallest perfect square less than 65 is 64, and the largest perfect square greater than 65 is 81. √65 falls somewhere between 8 and 9.

Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula, (65 - 64) / (81 - 64) ≈ 1 / 17 ≈ 0.059 Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number which is 8 + 0.059 = 8.059, so the square root of 65 is approximately 8.06.

• 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, 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, it is always the positive square root that has more prominence due to its uses in the real world. This is known as the principal square root.

• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 64 is a perfect square because it is 8 squared (8 x 8).