Square Root of 20000

The square root is the inverse of the square of the number. 20000 is not a perfect square. The square root of 20000 is expressed in both radical and exponential form. In the radical form, it is expressed as √20000, whereas (20000)^(1/2) in the exponential form. √20000 ≈ 141.421, 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 20000 is broken down into its prime factors.

Step 1: Finding the prime factors of 20000 Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5: 2⁴ x 5⁵

Step 2: Now we found out the prime factors of 20000. The second step is to make pairs of those prime factors. Since 20000 is not a perfect square, therefore the digits of the number can’t be grouped in a perfect pair.

Therefore, calculating √20000 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 20000, we need to group it as 000 and 20.

Step 2: Now we need to find n whose square is 20. We can say n as ‘4’ because 4 × 4 is lesser than or equal to 20. Now the quotient is 4, and after subtracting 16 from 20, the remainder is 4.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 4 + 4 to get 8, which will be our new divisor.

Step 4: The new divisor will be 8n. Now we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 400. Let us consider n as 5, now 8 × 5 × 5 = 400.

Step 6: Subtract 400 from 400, the difference is 0, and the quotient is 45.

Step 7: Since the dividend is zero, we have our quotient as 141.42. So the square root of √20000 ≈ 141.42

The approximation method is another method for finding the 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 20000 using the approximation method.

Step 1: Now we have to find the closest perfect square of √20000. The smallest perfect square of 20000 is 19600 (140²) and the largest perfect square of 20000 is 20449 (143²). √20000 falls somewhere between 140 and 143.

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 (20000 - 19600) ÷ (20449 - 19600) ≈ 0.42 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 140 + 0.42 ≈ 140.42, so the square root of 20000 is approximately 141.42.

• Square root: A square root is the inverse of a square. 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.

• Perfect square: A perfect square is a number that is the square of an integer. For example: 16 is a perfect square because it is 4².

• Long division method: A mathematical procedure used to find the square root of non-perfect squares through a step-by-step division process.

• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3^2.