Square Root of 488

The square root is the inverse of the square of the number. 488 is not a perfect square. The square root of 488 is expressed in both radical and exponential form.

In the radical form, it is expressed as √488, whereas (488^{1/2}) in the exponential form. √488 ≈ 22.0907, 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 488 is broken down into its prime factors.

Step 1: Finding the prime factors of 488

Breaking it down, we get 2 x 2 x 2 x 61.

Step 2: Now we found out the prime factors of 488. The second step is to make pairs of those prime factors. Since 488 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 488 using prime factorization is not straightforward.

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 488, we need to group it as 88 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 lesser than or equal to 4. Now the quotient is 2, and after subtracting 4-4, the remainder is 0.

Step 3: Now let us bring down 88, which is the new dividend. Add the old divisor with the same number 2 + 2 to 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, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 88. Let us consider n as 2, now 4 x 2 x 2 = 16.

Step 6: Subtract 88 from 48, and the difference is 40.

Step 7: Since the dividend is not less than the previous 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 4000.

Step 8: Now we need to find the new divisor, which is 209 because 4409 x 9 = 39681.

Step 9: Subtracting 39681 from 40000, we get the result 319.

Step 10: Now the quotient is 22.09.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue till the remainder is zero.

So the square root of √488 ≈ 22.09

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 488 using the approximation method.

Step 1: Now we have to find the closest perfect square of √488. The smallest perfect square less than 488 is 484, and the largest perfect square greater than 488 is 529. √488 falls somewhere between 22 and 23.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (488 - 484) / (529-484) ≈ 0.089.

Using the formula, we identified the decimal approximation of our square root. The next step is adding the value we got initially to the decimal number, which is 22 + 0.0907 ≈ 22.0907.

So the square root of 488 is approximately 22.0907.

• Square root: A square root is the inverse operation of squaring a number. For example, \(4^2 = 16\), and the inverse operation is finding the square root, \( \sqrt{16} = 4\).
   
• 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.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is usually taken in practical applications, which is known as the principal square root.
   
• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 488 is \(2^3 \times 61\).
   
• Approximation: Estimating a value that is close to the actual value, often used for non-perfect squares like 488.