Square Root of 2.15

The square root is the inverse of the square of the number. 2.15 is not a perfect square. The square root of 2.15 is expressed in both radical and exponential forms. In the radical form, it is expressed as √2.15, whereas (2.15)¹/² in the exponential form. √2.15 ≈ 1.4641, 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:

• Long division method
• Approximation method

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 consider 2.15 as 215/100. Group the numbers from right to left. In the case of 215, group it as 2 and 15.

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

Step 3: Now let us bring down the pair 15, making it 115. Add the old divisor with the same number 1 + 1 to 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: Find 2n × n ≤ 115. Let us consider n as 5, now 25 × 5 = 125, which is too large, so try n = 4, making 24 × 4 = 96.

Step 6: Subtract 96 from 115; the difference is 19, and the quotient is 1.4.

Step 7: 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 1900.

Step 8: Now we need to find the new divisor, which is 29, because 294 × 6 = 1764.

Step 9: Subtracting 1764 from 1900, we get the result 136.

Step 10: Now the quotient is 1.46.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √2.15 is approximately 1.4641.

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 2.15 using the approximation method.

Step 1: Find the closest perfect square to √2.15. The smallest perfect square less than 2.15 is 1, and the largest perfect square greater than 2.15 is 4. √2.15 falls somewhere between 1 and 2.

Step 2: Now apply the formula that is: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Using the formula (2.15 - 1) / (4 - 1) = 1.15 / 3 = 0.3833. Using this formula, we approximate the decimal part of our square root. The next step is adding the value we got initially to the decimal approximation, which is 1 + 0.3833 ≈ 1.3833. By further refinement, we find √2.15 ≈ 1.4641.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2.
   
• Irrational number: An irrational number is a number that cannot be written as a simple fraction - the ratio of two integers. Examples include √2, √3, and π.
   
• Approximation: An approximation is a value or quantity that is nearly but not exactly correct. Approximations are often used in calculating square roots of non-perfect squares.
   
• Decimal: A decimal is a fraction written in a special form. Instead of writing 1/2, you can express it as 0.5. Decimals are another way to show fractions.
   
• Long Division Method: A technique used to divide numbers to find the quotient, and in terms of square roots, to find the square root of a number with more precision than basic methods.