Square Root of 50000

The square root is the inverse of the square of the number. 50000 is not a perfect square. The square root of 50000 is expressed in both radical and exponential form. In radical form, it is expressed as √50000, whereas (50000)^(1/2) in exponential form. √50000 = 223.6068, 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 50000 is broken down into its prime factors:

Step 1: Finding the prime factors of 50000 Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5: 2^4 x 5^5

Step 2: Now that we found the prime factors of 50000, the second step is to make pairs of those prime factors. Since 50000 is not a perfect square, therefore the digits of the number can’t be grouped into pairs without a remainder.

Therefore, calculating 50000 using prime factorization requires further steps beyond simple pairing.

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 50000, we need to group it as 500 and 00.

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

Step 3: Now let us bring down the next pair of digits, which is 00, making the new dividend 100. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: The new divisor is 4n. We need to find n such that 4n x n ≤ 100. Let's consider n as 2, now 42 x 2 = 84.

Step 5: Subtract 84 from 100, the difference is 16, and the quotient is 22.

Step 6: Bring down the next pair of digits, 00, making the new dividend 1600.

Step 7: Now we find the new divisor, which is 444, because 444 x 3 = 1332.

Step 8: Subtracting 1332 from 1600 gives the result 268.

Step 9: 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 26800.

Step 10: Continue doing these steps until we get the desired precision.

The result will be approximately 223.6068.

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 50000 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √50000. The smallest perfect square less than 50000 is 49000 and the largest perfect square greater than 50000 is 50625. √50000 falls somewhere between 220 and 225.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Going by the formula: (50000 - 49000) ÷ (50625 - 49000) = 1000 / 1625 ≈ 0.615 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially (220) to the decimal number: 220 + 0.615 = 220.615.

So the square root of 50000 is approximately 223.6068 after further refinement.

• Square root: A square root is the inverse of a square. 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, it is always a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 50000 is 2^4 x 5^5.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.