Square Root of 588

The square root is the inverse of the square of the number. 588 is not a perfect square. The square root of 588 is expressed in both radical and exponential form. In the radical form, it is expressed as √588, whereas (588)^(1/2) in the exponential form. √588 ≈ 24.2487, 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 588 is broken down into its prime factors.

Step 1: Finding the prime factors of 588 Breaking it down, we get 2 x 2 x 3 x 7 x 7: 2^2 x 3 x 7^2.

Step 2: Now we found out the prime factors of 588. The second step is to make pairs of those prime factors. Since 588 is not a perfect square, the digits of the number can’t be grouped in perfect pairs for a square root without leaving a remainder.

Therefore, calculating 588 using prime factorization results in an approximate 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 588, we group it as 88 and 5.

Step 2: Now we need to find n whose square is ≤ 5. We can say n as ‘2’ because 2 x 2 = 4 is lesser than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 88, making the new dividend 188. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n ≤ 188. Let’s consider n as 4, then 44 x 4 = 176.

Step 5: Subtract 176 from 188; the difference is 12, and the quotient is 24.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point, allowing us to add two zeros to the dividend. Now the new dividend is 1200.

Step 7: The new divisor should be 244 because 244 x 4 = 976, which is less than 1200.

Step 8: Subtracting 976 from 1200 gives us a remainder of 224.

Step 9: Continue this process until we reach a desired level of decimal precision.

The approximate square root of 588 is 24.248.

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 588 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √588. The closest perfect squares are 576 (24^2) and 625 (25^2), so √588 falls somewhere between 24 and 25.

Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (588 - 576) / (625 - 576) = 12 / 49 ≈ 0.245. Adding this to 24 gives us 24 + 0.245 = 24.245, so the square root of 588 is approximately 24.245.

• Square root: A square root is the inverse of a square. For 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, the positive square root is more prominent due to its uses in the real world. This is why it is also known as the principal square root.

• Factors: Factors are numbers that can be multiplied together to get another number. For example, factors of 588 are 1, 2, 3, 4, 6, 7, etc.

• Approximation: An approximation is a value or number that is close to the exact value but not exact. For example, the approximate square root of 588 is 24.2487.