Square Root of 490

The square root is the inverse of the square of the number. 490 is not a perfect square. The square root of 490 is expressed in both radical and exponential forms.

In the radical form, it is expressed as √490, whereas (490)^(1/2) in the exponential form. √490 = 22.13594362, 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 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 490 is broken down into its prime factors:

Step 1: Finding the prime factors of 490

Breaking it down, we get 2 x 5 x 7 x 7: 2^1 x 5^1 x 7^2

Step 2: Now we found the prime factors of 490. Since 490 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 490 using prime factorization is not straightforward for finding an exact square root.

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 490, we need to group it as 90 and 4.

Step 2: Now we need to find n whose square is 4. We can say n as ‘2’ because 2 x 2 is less than or equal to 4. Now the quotient is 2 after subtracting 4-4 the remainder is 0.

Step 3: Now let us bring down 90 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 the sum of the dividend and quotient. Now we get 4n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 90. Let us consider n as 2. Now 4 x 2 x 2 = 8 x 2 = 16

Step 6: Subtract 90 from 16, the difference is 74, and the quotient is 22

Step 7: 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 7400.

Step 8: Now we need to find the new divisor that is 445 because 445 x 5 = 2225

Step 9: Subtracting 2225 from 7400, we get the result 5175.

Step 10: Now the quotient is 22.1

Step 11: 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 √490 is approximately 22.13.

The approximation method is another method for finding 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 490 using the approximation method.

Step 1: Now we have to find the closest perfect square of √490. The smallest perfect square less than 490 is 484 and the largest perfect square greater than 490 is 529. √490 falls somewhere between 22 and 23.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (490 - 484) / (529 - 484) = 6/45 ≈ 0.1333

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 22 + 0.1333 = 22.1333, so the square root of 490 is approximately 22.13.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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 q is not equal to zero and p and q are integers.
   
• Exponent: An exponent represents the number of times a base number is multiplied by itself. For example: 2^3 = 2 x 2 x 2 = 8.
   
• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 50 is 2 x 5 x 5.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example: 9 is a perfect square because it is 3^2.