Square Root of 749

The square root is the inverse of the square of the number. 749 is not a perfect square. The square root of 749 is expressed in both radical and exponential form. In the radical form, it is expressed as √749, whereas (749)¹/₂ in the exponential form. √749 ≈ 27.370, 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, it 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 749 is broken down into its prime factors.

Step 1: Finding the prime factors of 749 Breaking it down, we get 7 x 107, where both 7 and 107 are prime numbers.

Step 2: Since 749 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 749 using prime factorization is not feasible 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 749, we need to group it as 49 and 7.

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

Step 3: Bring down 49, making the new dividend 349. Add the old divisor (2) with the same number (2) to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. Now, find n such that 4n x n ≤ 349. Let us consider n as 8, now 48 x 8 = 384.

Step 5: Since 349 is less than 384, use a smaller n. Try n = 7, 47 x 7 = 329.

Step 6: Subtract 329 from 349, the difference is 20, and the quotient is 27.

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 2000.

Step 8: Find the new divisor by adding 7 to 47, making it 54. Find n such that 54n x n ≤ 2000.

Step 9: Continue the process until a satisfactory precision is achieved.

The square root of √749 is approximately 27.370.

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 749 using the approximation method.

Step 1: We have to find the closest perfect square of √749.

The smallest perfect square less than 749 is 729 (27²) and the largest perfect square greater than 749 is 784 (28²). √749 falls somewhere between 27 and 28.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula: (749 - 729) ÷ (784 - 729) = 20 ÷ 55 ≈ 0.364.

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: 27 + 0.364 ≈ 27.364, so the square root of 749 is approximately 27.364.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16 and the inverse of the square is the square root, which is √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a fraction, i.e., 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 more commonly used due to its applications in the real world. This is known as the principal square root.

• Decimal: If a number has a whole number and a fractional part, then it is called a decimal. For example: 27.37, 8.65, and 9.42 are decimals.

• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 749 is 7 × 107.