Square Root of 732

The square root is the inverse of the square of the number. 732 is not a perfect square. The square root of 732 is expressed in both radical and exponential form. In the radical form, it is expressed as √732, whereas in the exponential form as (732)^(1/2). √732 ≈ 27.0555, 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 732, 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 732 is broken down into its prime factors:

Step 1: Finding the prime factors of 732

Breaking it down, we get 2 × 2 × 3 × 61: 2^2 × 3 × 61

Step 2: Now we have found the prime factors of 732. The next step is to make pairs of those prime factors. Since 732 is not a perfect square, the digits of the number can’t be grouped in pairs to find an exact square root. Therefore, calculating √732 using prime factorization alone is not feasible.

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 732, we need to group it as 32 and 7.

Step 2: Now we need to find n whose square is ≤ 7. We can say n is '2' because 2 × 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 32, making the new dividend 332. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the quotient and twice the previous quotient. We get 4n as the new divisor. We need to find the value of n such that 4n × n ≤ 332.

Step 5: Consider n as 8, now 48 × 8 = 384, which is less than 332. We need to try a smaller number.

Step 6: Trying n as 7, 47 × 7 = 329, which fits. So we subtract 329 from 332, leaving a remainder of 3, and the quotient is 27.

Step 7: 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, making it 300.

Step 8: Now find the new divisor, which is 540 because 540 × 0.5 = 270, which is less than 300.

Step 9: Subtracting 270 from 300 gives a remainder of 30.

Step 10: Continue these steps until we get two numbers after the decimal point. If there is no remainder, continue until the remainder is zero.

So the square root of √732 ≈ 27.0555.

The approximation method is an easy way to find the square root of a given number. Now let us learn how to find the square root of 732 using the approximation method.

Step 1: Find the closest perfect squares to √732. The smallest perfect square less than 732 is 729, and the largest perfect square greater than 732 is 784. √732 falls between 27 and 28.

Step 2: Now apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (732 - 729) ÷ (784 - 729) = 3 ÷ 55 ≈ 0.055 Add the value obtained to the lower bound: 27 + 0.055 ≈ 27.055.

So the square root of 732 is approximately 27.055.

• Square root: A square root is the inverse of a square. For example, 5² = 25 and the inverse of the square is the square root, which is √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.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 144 is a perfect square since it is 12².

• Long division method: A method used to find the square root of numbers, especially when they are not perfect squares, through systematic division.

• Approximation method: A technique used to estimate the square root of a number by comparing it to the nearest perfect squares and using interpolation.