Square Root of 494209

The square root is the inverse of the square of the number. 494209 is a perfect square. The square root of 494209 is expressed in both radical and exponential form. In the radical form, it is expressed as √494209, whereas (494209)^(1/2) in the exponential form. √494209 = 703, which is a rational number because it can 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. For non-perfect square numbers, methods like 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 494209 is broken down into its prime factors.

Step 1: Finding the prime factors of 494209 Breaking it down, we get 7 x 7 x 7 x 7 x 7 x 7: 7^6

Step 2: Now we found out the prime factors of 494209. The second step is to make pairs of those prime factors. Since 494209 is a perfect square, the digits of the number can be grouped in pairs. Therefore, calculating 494209 using prime factorization is possible, and the square root is 703.

The long division method is particularly used for both perfect and 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 494209, we need to group it as 09, 42, and 49.

Step 2: Now we need to find n whose square is less than or equal to 49. We can say n is ‘7’ because 7 x 7 = 49. Now the quotient is 7, and after subtracting 49-49, the remainder is 0.

Step 3: Bring down 42, which is the new dividend. Add the old divisor with the same number: 7 + 7 = 14, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and the quotient. Now we get 14n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 14n × n ≤ 42. Let us consider n as 0, now 140 x 0 = 0.

Step 6: Subtract 42 from 0; the difference is 42, and the quotient is 70.

Step 7: Bring down 09 to make the new dividend 4209. Add the decimal point to add two zeroes to the dividend.

Step 8: Find the new divisor that is 1406 because 1406 x 3 = 4218, which is closest to 4209.

Step 9: Subtracting 4218 from 4209 gives a negative remainder, so adjust by using 1403 x 3 = 4209.

Step 10: Now the quotient is 703. So the square root of √494209 is 703.

Since 494209 is a perfect square, the approximation method is not required. However, if it were needed, the closest perfect squares would be 490000 (700) and 504100 (710), placing 494209 around the midpoint, confirming √494209 is exactly 703.

• Square root: A square root is the inverse of a square. Example: 703^2 = 494209, and the inverse of the square is the square root, that is, √494209 = 703.

• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero, and p and q are integers.

• Prime factorization: The expression of a number as the product of its prime numbers. Example: 494209 = 7^6.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 494209 is a perfect square because 703^2 = 494209.

• Long division method: A method used to find the square root of a number by dividing the number into pairs of digits starting from the right.