Square Root of 608

The square root is the inverse of the square of the number. 608 is not a perfect square. The square root of 608 is expressed in both radical and exponential form. In the radical form, it is expressed as √608, whereas (608)^(1/2) in the exponential form. √608 ≈ 24.657, 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 608 is broken down into its prime factors.

Step 1: Finding the prime factors of 608 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 19 = 2^5 x 19

Step 2: Now we found out the prime factors of 608. The second step is to make pairs of those prime factors. Since 608 is not a perfect square, the digits of the number can't be grouped in pairs completely. Therefore, calculating 608 using prime factorization leaves us with a residual factor.

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 608, we need to group it as 08 and 6.

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

Step 3: Now let us bring down 08, 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 value of n.

Step 5: The next step is finding 4n x n ≤ 208. Let us consider n as 5, now 45 x 5 = 225, which is more than 208, so we consider n as 4.

Step 6: 44 x 4 = 176. Subtracting 176 from 208, the difference is 32, and the quotient is 24.

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 3200.

Step 8: Now we need to find the new divisor, which is 4n. Let's assume n as 6, 496 x 6 = 2976.

Step 9: Subtracting 2976 from 3200, we get the result 224.

Step 10: Now the quotient is 24.6

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there is no decimal value, continue until the remainder is zero. So the square root of √608 is approximately 24.66.

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 608 using the approximation method.

Step 1: Now we have to find the closest perfect square of √608. The smallest perfect square less than 608 is 576 (24^2), and the largest perfect square greater than 608 is 625 (25^2). √608 falls somewhere between 24 and 25.

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 (608 - 576) ÷ (625 - 576) = 32 / 49 ≈ 0.653 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 24 + 0.653 ≈ 24.653. So the square root of 608 is approximately 24.653.

• 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.

• Principal square root: A number has both positive and negative square roots; however, it is always a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Prime factorization: Expressing a number as a product of its prime factors is called prime factorization.

• 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.