Square Root of 703

The square root is the inverse of the square of the number. 703 is not a perfect square. The square root of 703 is expressed in both radical and exponential forms. In the radical form, it is expressed as √703, whereas (703)^(1/2) in the exponential form. √703 ≈ 26.5141, 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 used for non-perfect square numbers where 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 703 is broken down into its prime factors.

Step 1: Finding the prime factors of 703. Breaking it down, we get 19 × 37: 19^1 × 37^1.

Step 2: Now we found out the prime factors of 703. The second step is to make pairs of those prime factors. Since 703 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 703 using prime factorization is impossible.

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 703, we need to group it as 03 and 7.

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

Step 3: Bring down 03, making it 303, which is the new dividend. Add the old divisor with the same number, 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n × n is less than or equal to 303. Let's consider n as 6, then 46 × 6 = 276.

Step 5: Subtract 276 from 303, and the difference is 27, and the quotient is 26.

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 2700.

Step 7: Now we need to find the new divisor, which will be 532 because 532 × 5 = 2660.

Step 8: Subtracting 2660 from 2700, we get 40.

Step 9: Continue doing these steps until we get two numbers after the decimal point.

So the approximate square root of √703 ≈ 26.5141.

Approximation method is another method for finding the 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 703 using the approximation method.

Step 1: Now we have to find the closest perfect square of √703. The smallest perfect square less than 703 is 676, and the largest perfect square greater than 703 is 729. √703 falls somewhere between 26 and 27.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (703 - 676) / (729 - 676) = 27/53 ≈ 0.509. Adding the integer part, 26, to the decimal, 0.509, we get 26.509. Thus, the square root of 703 is approximately 26.509.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that 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, it is always the positive square root that is more commonly used due to its relevance in real-world applications. That is the reason it is also known as the principal square root.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4².

• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime numbers.