Square Root of 755

The square root is the inverse of squaring a number. 755 is not a perfect square. The square root of 755 is expressed in both radical and exponential forms. In radical form, it is expressed as √755, whereas in exponential form it is expressed as (755)^(1/2). √755 ≈ 27.477, 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, methods like 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 755 is broken down into its prime factors:

Step 1: Finding the prime factors of 755 Breaking it down, we get 5 x 151. These are both prime numbers.

Step 2: Since 755 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating √755 using prime factorization is not straightforward.

The long division method is particularly useful for non-perfect square numbers. Let's learn how to find the square root using the long division method, step by step.

Step 1: Begin by grouping the digits of 755 from right to left, which gives us 55 and 7.

Step 2: Find n whose square is ≤ 7. We can say n = 2 because 2 x 2 = 4, which is less than or equal to 7. The quotient is 2 after subtracting 4 from 7, leaving a remainder of 3.

Step 3: Bring down the next pair of digits, 55, making the new dividend 355. Double the current quotient (2) to get 4, which will be part of the new divisor.

Step 4: The new divisor will be 4n. We need to find n such that 4n x n ≤ 355. Let n = 7, giving us 47 x 7 = 329.

Step 5: Subtract 329 from 355, getting a remainder of 26. The quotient now is 27.

Step 6: 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, making it 2600.

Step 7: Find a new digit for n such that 540x x x ≤ 2600. We find x = 4 works, as 544 x 4 = 2176.

Step 8: Subtract 2176 from 2600, resulting in a remainder of 424. Continue the process to get more decimal places if necessary.

So the square root of √755 is approximately 27.477.

The approximation method is another method for finding square roots, providing an easy way to estimate the square root of a given number. Let's learn how to approximate the square root of 755.

Step 1: Find the closest perfect squares to √755.

The smaller perfect square closest to 755 is 729, and the larger one is 784. √755 falls somewhere between 27 and 28.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Using the formula, (755 - 729) / (784 - 729) ≈ 0.477. Adding this decimal to the smaller perfect square root gives us 27 + 0.477 = 27.477.

Therefore, the square root of 755 is approximately 27.477.

• Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 16, and the square root is √16 = 4.

• Irrational number: An irrational number cannot be written in the form of p/q, where q is not zero and p and q are integers.

• Real number: A real number can be a rational or irrational number and is often represented on a number line.

• Approximation: An approximation is an estimated value that is close to the actual value.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.