Square Root of 559

The square root is the inverse of the square of the number. 559 is not a perfect square. The square root of 559 is expressed in both radical and exponential form. In the radical form, it is expressed as √559, whereas (559)^(1/2) in the exponential form. √559 ≈ 23.629, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 559 is broken down into its prime factors.

Step 1: Finding the prime factors of 559 Breaking it down, we get 13 x 43: 13¹ x 43¹

Step 2: Now we found out the prime factors of 559. The second step is to make pairs of those prime factors. Since 559 is not a perfect square, grouping the digits in pairs is not possible.

Therefore, calculating √559 using prime factorization is impractical.

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 559, we need to group it as 59 and 5.

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

Step 3: Now let us bring down 59, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 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 ≤ 159. Let us consider n as 3; now 43 x 3 = 129.

Step 6: Subtract 129 from 159; the difference is 30, and the quotient is 23.

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 3000.

Step 8: Now we need to find the new divisor that is 7 because 473 x 7 = 3311. Step 9: Subtracting 3311 from 3000, we get the result -311.

Step 10: Now the quotient is 23.6.

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 √559 ≈ 23.63.

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 559 using the approximation method.

Step 1: Now we have to find the closest perfect square of √559. The smallest perfect square less than 559 is 529, and the largest perfect square greater than 559 is 576. √559 falls somewhere between 23 and 24.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (559 - 529) ÷ (576 - 529) = 0.638.

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 23 + 0.638 = 23.638.

So the square root of 559 is approximately 23.638.

• Square root: A square root is the inverse of a square. For example, 4² = 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.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 559 is 13 x 43.

• Long division method: A method used to find the square root of non-perfect squares through division, involving a series of steps to approximate the square root.

• Decimal: A number that has a whole number and a fractional part separated by a decimal point, for example, 23.629.