Square Root of 560

The square root is the inverse of the square of a number. 560 is not a perfect square. The square root of 560 is expressed in both radical and exponential form. In the radical form, it is expressed as √560, whereas (560)^(1/2) in the exponential form. √560 ≈ 23.6643, 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 560 is broken down into its prime factors.

Step 1: Finding the prime factors of 560 Breaking it down, we get 2 × 2 × 2 × 2 × 5 × 7: 2⁴ × 5¹ × 7¹

Step 2: Now we found the prime factors of 560. The second step is to make pairs of those prime factors. Since 560 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 560 using prime factorization is impossible.

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 560, we need to group it as 60 and 5.

Step 2: Now we need to find n whose square is 5. We can say n as ‘2’ because 2 × 2 = 4, which is less 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 60, 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 × n ≤ 160. Let us consider n as 3, now 4 × 3 × 3 = 144.

Step 6: Subtract 144 from 160, the difference is 16, 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 1600.

Step 8: Now we need to find the new divisor, which is 466 because 466 × 3 = 1398.

Step 9: Subtracting 1398 from 1600, we get the result 202.

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 √560 is approximately 23.66.

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 560 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √560. The smallest perfect square less than 560 is 529, and the largest perfect square greater than 560 is 576. √560 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 (560 - 529) ÷ (576 - 529) = 31 ÷ 47 ≈ 0.66. 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 + 0.66 = 23.66.

So the square root of 560 is approximately 23.66.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 × 4 = 16.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction; it has a non-repeating and non-terminating decimal expansion.

• Radical expression: A radical expression is an expression that includes a square root, cube root, or higher order roots.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups of digits.

• Approximation method: A method used to estimate the square root of a number by finding the closest perfect squares around the number.