Square Root of 1288

The square root is the inverse of the square of a number. 1288 is not a perfect square. The square root of 1288 is expressed in both radical and exponential form. In radical form, it is expressed as √1288, whereas 1288^(1/2) in exponential form. √1288 ≈ 35.877, 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, for non-perfect square numbers like 1288, 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 1288 is broken down into its prime factors.

Step 1: Finding the prime factors of 1288

Breaking it down, we get 2 x 2 x 2 x 7 x 23: 2^3 x 7 x 23

Step 2: Now we have found the prime factors of 1288. The second step is to make pairs of those prime factors. Since 1288 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 1288 using prime factorization alone is incomplete.

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, we need to group the numbers from right to left. In the case of 1288, we need to group it as 28 and 12.

Step 2: Now we need to find n whose square is ≤ 12. We can say n as '3' because 3 x 3 = 9 is lesser than or equal to 12. Now the quotient is 3, and after subtracting 9 from 12, the remainder is 3.

Step 3: Bring down 88, making it the new dividend. Add the old divisor with the same number 3 + 3, which gives us 6 as the beginning of our new divisor.

Step 4: Find a digit, say m, such that 6m x m is ≤ 388. The closest we get is 64 x 4 = 256.

Step 5: Subtract 256 from 388, yielding a remainder of 132.

Step 6: Since the dividend is less than the divisor, we add a decimal point and bring down two zeroes. The new dividend is 13200.

Step 7: Repeat these steps until we get two numbers after the decimal point. The process continues to yield more precise decimal places.

So the square root of √1288 is approximately 35.877.

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 1288 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 1288. The smallest perfect square less than 1288 is 1225, and the largest perfect square greater than 1288 is 1369. √1288 falls somewhere between 35 and 37.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying the formula: (1288 - 1225) / (1369 - 1225) = 63 / 144 ≈ 0.4375. Using the formula, we identified the decimal point of our square root. Adding this to the initial value: 35 + 0.4375 ≈ 35.877, so the square root of 1288 is approximately 35.877.

• 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 used in practical applications. This is known as the principal square root.

• Long division method: A method used to find the square root of non-perfect squares by a series of steps involving division and subtraction.

• Prime factorization: Breaking down a number into its basic prime factors, which can help in simplifying radical expressions.