Square Root of 739

The square root is the inverse of the square of a number. 739 is not a perfect square. The square root of 739 is expressed in both radical and exponential form. In radical form, it is expressed as √739, whereas (739)^(1/2) is the exponential form. √739 ≈ 27.18935, which is an irrational number because it cannot be expressed as 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 long division and approximation 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 739 is broken down into its prime factors:

Step 1: Finding the prime factors of 739. 739 is a prime number itself, so its only factors are 1 and 739. Since 739 is not a perfect square, calculating √739 using prime factorization in pairs is not feasible.

The long division method is particularly useful 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. For 739, group it as 39 and 7.

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

Step 3: Bring down 39, making the new dividend 339. Add the old divisor with the same number, 2 + 2 = 4, to get a new divisor.

Step 4: Now find the largest n such that 4n × n ≤ 339. Let’s consider n as 8. Now, 48 × 8 = 384, which is greater than 339. Trying n as 7, we get 47 × 7 = 329, which is less than 339.

Step 5: Subtract 329 from 339, the difference is 10, and the quotient is 27.

Step 6: Since the dividend is less than the divisor, we add a decimal point and two zeroes to the dividend. The new dividend is 1000.

Step 7: Find the new divisor. The divisor is now 274, and we try n as 3, since 2743 × 3 = 8229. Continue these steps until we achieve the desired precision.

The square root of √739 is approximately 27.189.

The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Let's learn how to find the square root of 739 using the approximation method.

Step 1: Find the closest perfect squares of √739. The smallest perfect square below 739 is 729 (since √729 = 27), and the largest perfect square nearby is 784 (since √784 = 28). Thus, √739 falls between 27 and 28.

Step 2: Apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula, (739 - 729) ÷ (784 - 729) = 10 ÷ 55 ≈ 0.1818. Add the initial value to the decimal number: 27 + 0.1818 ≈ 27.1818. Thus, the square root of 739 is approximately 27.1818.

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

• Irrational number: An irrational number 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 often used in real-world applications, known as the principal square root.

• Prime number: A prime number has exactly two distinct positive divisors: 1 and itself. For example, 739 is a prime number.

• Decimal: A decimal number contains a whole number and a fractional part separated by a decimal point. For example, 7.86, 8.65, and 9.42 are decimals.