Square Root of 62

The square root is the inverse of the square of the number. 62 is not a perfect square. The square root of 62 is expressed in both radical and exponential form. In the radical form, it is expressed as √62, whereas (62)^(1/2) in the exponential form.

√62 ≈ 7.874, 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 long-division and approximation methods 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 62 is broken down into its prime factors:

Step 1: Finding the prime factors of 62 Breaking it down, we get 2 x 31: 2¹ x 31¹

Step 2: Now we found out the prime factors of 62. The second step is to make pairs of those prime factors. Since 62 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 62 using prime factorization is impossible.

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 62, we need to group it as 62.

Step 2: Now we need to find n whose square is less than or equal to 62. We can approximate n as 7 because 7 x 7 = 49 is lesser than 62. Now the quotient is 7 after subtracting 49 from 62, the remainder is 13.

Step 3: Add a decimal point to the quotient and bring down 00 to make the remainder 1300.

Step 4: Double the quotient (ignoring the decimal) to get a new divisor, which is 14. Now we get 14n as the new divisor. We need to find the value of n.

Step 5: Find 14n x n ≤ 1300. Let us consider n as 9, now 149 x 9 = 1341, which is too large. Try n as 8, now 148 x 8 = 1184.

Step 6: Subtract 1184 from 1300, the difference is 116. The quotient is now 7.8.

Step 7: Add two zeroes to the remainder for further approximation. Now the new remainder is 11600.

Step 8: Continue the process until the desired precision is achieved, usually two decimal places.

So the square root of √62 ≈ 7.87.

The approximation method is another method for finding square roots, and 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 62 using the approximation method.

Step 1: Find the closest perfect squares around 62. The smallest perfect square less than 62 is 49 (7²), and the largest perfect square greater than 62 is 64 (8²). √62 falls somewhere between 7 and 8.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (62 - 49) / (64 - 49) = 13 / 15 ≈ 0.87 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 7 + 0.87 = 7.87, so the square root of 62 is approximately 7.87.

• Square root: A square root is the inverse of a square. For example, 7^2 = 49, and the inverse of the square is the square root, which is √49 = 7.

• 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 relevance in real-world applications. This is known as the principal square root.

• Prime factorization: Prime factorization involves expressing a number as the product of its prime factors. For example, the prime factorization of 62 is 2 x 31.

• Long division method: The long division method is a technique used to find the square root of a non-perfect square number through a series of division and approximation steps.