Square Root of 15600

The square root is the inverse of the square of the number. 15600 is not a perfect square. The square root of 15600 is expressed in both radical and exponential form. In the radical form, it is expressed as √15600, whereas (15600)^(1/2) in the exponential form. √15600 ≈ 124.899, 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 15600 is broken down into its prime factors.

Step 1: Finding the prime factors of 15600 Breaking it down, we get 2 × 2 × 2 × 2 × 3 × 5 × 5 × 13: 2⁴ × 3¹ × 5² × 13¹

Step 2: Now that we have found the prime factors of 15600, the next step is to make pairs of those prime factors. Since 15600 is not a perfect square, the digits of the number can’t be grouped entirely into pairs.

Therefore, calculating 15600 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 15600, we need to group it as 00, 56, and 15.

Step 2: Now, find n whose square is less than or equal to 15. We can say n is ‘3’ because 3 × 3 = 9, which is less than 15. Subtract 9 from 15 to get a remainder of 6. The quotient is 3.

Step 3: Bring down 56 to make the new dividend 656. Add the old divisor (3) to itself to get 6, which becomes the new divisor.

Step 4: Multiply the new divisor (6) by a number p such that 6p × p ≤ 656. The suitable value for p is 10 since 610 × 0 = 6100, which is less than 656.

Step 5: Subtract 6100 from 65600 to get a remainder of 4600.

Step 6: Bring down the next pair of zeroes to make the new dividend 460000. Add 10 to the previous divisor to make it 620.

Step 7: Find a new digit q such that 620q × q ≤ 460000. The suitable value for q is 7 since 6207 × 7 = 43449, which is less than 460000.

Step 8: Subtract 43449 from 460000 to get a remainder of 25551.

Step 9: Continue this process until you achieve the desired level of accuracy. The quotient obtained is approximately 124.899.

The approximation method is another way to find 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 15600 using the approximation method.

Step 1: First, find the closest perfect square to √15600. The closest perfect squares are 14400 (120²) and 16900 (130²). √15600 falls between 120 and 130.

Step 2: Now, apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Applying the formula, (15600 - 14400) ÷ (16900 - 14400) = 1200 ÷ 2500 = 0.48. Adding this to the lower limit, 120 + 0.48 ≈ 124.48, so the approximate square root of 15600 is 124.48.

• 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.

• Perfect square: A number that is the square of an integer. For example, 144 is a perfect square since it equals 12².

• Approximation method: A method used to estimate the square root of a number when it is not a perfect square, using nearby perfect squares for calculation.