Square Root of 757

The square root is the inverse of the square of the number. 757 is not a perfect square. The square root of 757 is expressed in both radical and exponential form. In the radical form, it is expressed as √757, whereas (757)^(1/2) in the exponential form. √757 ≈ 27.5098, 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 757, 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 757 is broken down into its prime factors:

Step 1: Finding the prime factors of 757 Breaking it down, we find that 757 is a product of primes: 757 = 7 x 7 x 11.

Step 2: Now that we found out the prime factors of 757, we realize that since 757 is not a perfect square, calculating √757 using prime factorization directly is not possible.

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, group the numbers from right to left. In the case of 757, group it as 57 and 7.

Step 2: Now find n whose square is less than or equal to 7. We can say n as '2' because 2 x 2 = 4 is less than 7. Now the quotient is 2, and after subtracting, the remainder is 3.

Step 3: Bring down 57 to make the new dividend 357. Double the previous quotient (2) to get 4, which will be part of the new divisor.

Step 4: Find n such that 4n x n is less than or equal to 357. Try n = 8, as 48 x 8 = 384 is too large, use n = 7, since 47 x 7 = 329.

Step 5: Subtract 329 from 357 to get 28. The quotient is now 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. Now the new dividend is 2800.

Step 7: Find the new divisor that is 547 because 547 x 5 = 2735.

Step 8: Subtract 2735 from 2800 to get 65.

Step 9: The quotient is now 27.5. Continue these steps until you reach the desired decimal precision.

So the square root of √757 is approximately 27.5098.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 757 using the approximation method.

Step 1: Find the closest perfect squares to √757.

The smallest perfect square less than 757 is 729 (27²) and the largest perfect square greater than 757 is 784 (28²). Thus, √757 falls between 27 and 28.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (757 - 729) / (784 - 729) = 28/55 ≈ 0.509. Adding this value to the nearest lower perfect square root, we have 27 + 0.509 = 27.509.

Thus, the square root of 757 is approximately 27.509.

• Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and the inverse of the square is the square root, so √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, but the positive square root is often used in real-world applications. This is why it is known as the principal square root.

• Perfect square: A perfect square is an integer that is the square of another integer. For example, 36 is a perfect square because 6² = 36.

• Approximation: In mathematics, an approximation is a value or quantity that is nearly but not exactly correct. For example, √757 ≈ 27.509 is an approximation.