Square Root of 758

The square root is the inverse of the square of the number. 758 is not a perfect square. The square root of 758 is expressed in both radical and exponential form. In radical form, it is expressed as √758, whereas (758)^(1/2) in exponential form. √758 ≈ 27.527, 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 typically used for non-perfect square numbers where long-division and approximation methods are applied. 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 758 is broken down into its prime factors.

Step 1: Finding the prime factors of 758 Breaking it down, we get 2 x 379: 2^1 x 379^1.

Step 2: Now we found the prime factors of 758. Since 758 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 758 using prime factorization is not straightforward for finding the square root.

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 758, we need to group it as 58 and 7.

Step 2: Now we need to find n whose square is ≤ 7. We can say n is ‘2’ because 2 x 2 = 4, which is less than 7. The quotient is 2 and the remainder is 3 after subtracting 4 from 7.

Step 3: Bring down 58, making the new dividend 358. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor is 4n. We need to find the value of n such that 4n x n ≤ 358. Let us consider n as 8, now 48 x 8 = 384, which is more than 358, so n = 7.

Step 5: 47 x 7 = 329. Subtract 329 from 358 to get 29.

Step 6: Since the dividend is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 2900.

Step 7: Find the new divisor. If n = 5, 545 x 5 = 2725.

Step 8: Subtract 2725 from 2900 to get 175.

Step 9: Continue these steps to get more decimal places.

The square root of √758 is approximately 27.53.

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 758 using the approximation method.

Step 1: Find the closest perfect squares around √758.

The smallest perfect square less than 758 is 729, and the largest perfect square greater than 758 is 784. √758 falls between 27 (√729) and 28 (√784).

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

Using the formula (758 - 729) / (784 - 729) ≈ 0.527. Add this to 27 to approximate the square root: 27 + 0.527 = 27.527.

• Square root: A square root is the inverse of squaring a number. Example: 5^2 = 25, and the inverse is the square root, √25 = 5.

• 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, the positive square root is used in most real-world applications. This is 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 it is 6^2.

• Long division method: A method used to find the square root of a non-perfect square number by dividing and averaging over several steps to get an approximate value.