Square Root of 1815

The square root is the inverse of the square of the number. 1815 is not a perfect square. The square root of 1815 is expressed in both radical and exponential form. In the radical form, it is expressed as √1815, whereas (1815)^(1/2) in the exponential form. √1815 ≈ 42.59798, 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 1815 is broken down into its prime factors.

Step 1: Finding the prime factors of 1815 Breaking it down, we get 3 x 5 x 11 x 11: 3^1 x 5^1 x 11^2

Step 2: Now we have found the prime factors of 1815. The next step is to make pairs of those prime factors. Since 1815 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating √1815 using prime factorization is not straightforward without approximation.

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 1815, we need to group it as 15 and 18.

Step 2: Now we need to find n whose square is 18. We can say n as ‘4’ because 4 x 4 = 16 is less than 18. Now the quotient is 4; after subtracting 18 - 16, the remainder is 2.

Step 3: Now let us bring down 15, which is the new dividend. Add the old divisor with the same number, 4 + 4, we get 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 215. Let us consider n as 2, now 8 x 2 x 2 = 32.

Step 6: Subtract 215 from 32; the difference is 183, and the quotient is 42.

Step 7: Since the dividend is greater 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 18300.

Step 8: Now we need to find the new divisor that is 849 because 849 x 9 = 7641.

Step 9: Subtracting 7641 from 18300, we get the result 10659.

Step 10: Now the quotient is 42.5.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero.

So, the square root of √1815 is approximately 42.60.

The approximation method is another way to find 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 1815 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1815. The smallest perfect square less than 1815 is 1764, and the largest perfect square greater than 1815 is 1849. √1815 falls somewhere between 42 and 43.

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

Going by the formula (1815 - 1764) ÷ (1849 - 1764) = 51 ÷ 85 = 0.6.

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 42 + 0.6 = 42.6, so the square root of 1815 is approximately 42.6.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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 usually the positive square root that is more prominent due to its uses in the real world. That is why it is also known as the principal square root.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.

• Long division method: This is a method used to find the square root of numbers, especially non-perfect squares, through a systematic, step-by-step division process.