Square Root of 82

The square root is the inverse of the square of the number. 82 is not a perfect square. The square root of 82 is expressed in both radical and exponential form.

In the radical form, it is expressed as √82, whereas (82)^(1/2) in the exponential form. √82 ≈ 9.05539, 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 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. Let's look at how 82 is broken down into its prime factors:

Step 1: Finding the prime factors of 82 Breaking it down, we get 2 x 41, which are both prime numbers.

Step 2: Now that we have found the prime factors of 82, we see that the number cannot be grouped into pairs.

Therefore, calculating 82 using prime factorization alone as a perfect square 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 82, we group it as 82.

Step 2: Now, we need to find n whose square is less than or equal to 82. We can say n as ‘9’ because 9 x 9 = 81.

Step 3: The quotient is 9, and after subtracting 81 from 82, the remainder is 1.

Step 4: Since the remainder is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the remainder. Now the new dividend is 100.

Step 5: The new divisor is 18 (double of 9), and we need to find the next digit of the quotient. We choose 5 because 185 x 5 = 925.

Step 6: Subtract 925 from 1000 to get 75.

Step 7: Continue doing these steps until we get two decimal places.

So the square root of √82 ≈ 9.05

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 82 using the approximation method.

Step 1: Now we have to find the closest perfect square of √82. The smallest perfect square less than 82 is 81, and the largest perfect square greater than 82 is 100. √82 falls between 9 and 10.

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

Using the formula: (82 - 81) / (100 - 81) = 1 / 19 ≈ 0.0526 Adding this to the base square root, 9 + 0.0526 = 9.0526,

so the square root of 82 is approximately 9.05.

• Square root: A square root is the inverse of squaring a number. Example: 3² = 9 and the inverse of the square is the square root that is √9 = 3.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction; its decimal representation is non-repeating and non-terminating.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 4, 9, and 16 are perfect squares.

• Radical: A radical is a symbol (√) that represents the root of a number.

• Approximation: An approximation is a value or number that is close to but not exactly equal to the exact value.