Square Root of 266.5

The square root is the inverse of the square of a number. 266.5 is not a perfect square. The square root of 266.5 is expressed in both radical and exponential form. In radical form, it is expressed as √266.5, whereas in exponential form, it is (266.5)^(1/2). √266.5 ≈ 16.32993, 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 266.5 can be broken down into its prime factors:

Step 1: Since 266.5 is not an integer, we cannot directly perform prime factorization. For approximation, consider 266 or 267.

Step 2: 266 = 2 x 133, and 267 = 3 x 89. Both 266 and 267 are not perfect squares, so calculating 266.5 using prime factorization is not feasible.

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 266.5, we need to group it as 66.5 and 2.

Step 2: Now we need to find n whose square is less than or equal to 2. We can say n as '1' because 1 x 1 is less than or equal to 2. Now the quotient is 1; after subtracting 2 - 1, the remainder is 1.

Step 3: Now let us bring down 66.5 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 2n. We need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 166.5. Let us consider n as 6, now 26 x 6 = 156

Step 6: Subtract 166.5 from 156; the difference is 10.5, and the quotient is 16.

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 1050.

Step 8: Now we need to find the new divisor that is 163 because 1633 x 3 = 489

Step 9: Subtracting 489 from 1050, we get the result 561.

Step 10: Now the quotient is 16.3

Step 11: 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 √266.5 is approximately 16.32993.

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 266.5 using the approximation method.

Step 1: Now we have to find the closest perfect square of √266.5. The smallest perfect square greater than 266.5 is 256, and the largest perfect square less than 266.5 is 289. √266.5 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 (266.5 - 256) ÷ (289 - 256) = 10.5 ÷ 33 ≈ 0.318 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 + 0.318 = 16.318, so the square root of 266.5 is approximately 16.318.

• 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. This is why it is also known as the principal square root.
   
• Long division method: A process used to find the square root of non-perfect squares by dividing the number into groups and performing division iteratively.
   
• Approximation method: A method used to estimate the square root of a non-perfect square by identifying the closest perfect squares and using linear interpolation to find a more precise value.