Square Root of 282

The square root is the inverse of the square of the number. 282 is not a perfect square. The square root of 282 is expressed in both radical and exponential form. In the radical form, it is expressed as √282, whereas (282)^(1/2) in the exponential form. √282 ≈ 16.794, 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 282 is broken down into its prime factors.

Step 1: Finding the prime factors of 282

Breaking it down, we get 2 x 3 x 47: 2^1 x 3^1 x 47^1

Step 2: Now we found out the prime factors of 282. The second step is to make pairs of those prime factors. Since 282 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 282 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 282, we need to group it as 82 and 2.

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

Step 3: Now let us bring down 82, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 182. Let us consider n as 8, now 28 x 8 = 224.

Step 6: Subtract 224 from 182, the difference is negative, so n must be less than 8. Try n = 7, now 27 x 7 = 189 which is still larger than 182.

Step 7: Try n = 6, now 26 x 6 = 156 which is less than 182.

Step 8: Subtract 156 from 182, the difference is 26, and the quotient is 16.

Step 9: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2600.

Step 10: Now we need to find the new divisor. Add the previous quotient, 16, to the divisor, 26, forming 272. Consider n as 9, 2729 x 9 = 24561, which is larger than 26000.

Step 11: Try n = 8, now 2728 x 8 = 21824, which is less than 26000.

Step 12: Subtract 21824 from 26000, the remainder is 4176, and the new quotient is 16.8.

Step 13: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value continue till the remainder is zero.

So the square root of √282 is approximately 16.79.

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 282 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √282. The smallest perfect square less than 282 is 256 (16^2), and the largest perfect square greater than 282 is 289 (17^2). √282 falls somewhere between 16 and 17.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (282 - 256) ÷ (289 - 256) = 26 / 33 ≈ 0.7879. 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 16 + 0.79 = 16.79, so the square root of 282 is approximately 16.79.

• 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; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Prime factorization: Prime factorization is expressing a number as a product of its prime numbers. For example, the prime factorization of 282 is 2 x 3 x 47.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.