Square Root of 673

The square root is the inverse of the square of the number. 673 is not a perfect square. The square root of 673 is expressed in both radical and exponential form. In the radical form, it is expressed as √673, whereas (673)¹/² in the exponential form. √673 ≈ 25.922, 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 673 is broken down into its prime factors.

Step 1: Finding the prime factors of 673 Since 673 is a prime number, it cannot be broken down further.

Therefore, calculating 673 using prime factorization is not applicable.

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 673, we need to group it as 73 and 6.

Step 2: Now we need to find n whose square is less than or equal to 6. We can say n is ‘2’ because 2 × 2 = 4, which is less than 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.

Step 3: Now let us bring down 73, which is the new dividend. Add the old divisor with the same number, 2 + 2, we get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the largest digit n such that 4n × n is less than or equal to 273. Let us consider n as 6. Now, 46 × 6 = 276, which is greater than 273. Trying n as 5, we get 45 × 5 = 225, which is less than 273.

Step 5: Subtract 225 from 273; the difference is 48, and the quotient is 25.

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 4800.

Step 7: Now, we need to calculate a new divisor 50n, such that 50n × n is less than or equal to 4800. Trying n as 9, we get 509 × 9 = 4581.

Step 8: Subtracting 4581 from 4800 gives the result 219. Step 9: Now the quotient is 25.9.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √673 is approximately 25.92.

The approximation method is another way to find 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 673 using the approximation method.

Step 1: Now we have to find the closest perfect square of √673. The smallest perfect square less than 673 is 625, and the largest perfect square more than 673 is 676. √673 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (673 - 625) ÷ (676 - 625) ≈ 0.94. 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, which is 25 + 0.94 = 25.94, so the square root of 673 is approximately 25.94.

• 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 is a number that cannot be written in the form of p/q, where p and q are integers, and q is not equal to zero.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 36 is a perfect square because it is 6².
   
• Prime number: A prime number is a natural number greater than 1 that is not divisible by any number other than 1 and itself. Example: 673 is a prime number.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.