Square Root of 90000

The square root is the inverse of the square of a number. 90000 is a perfect square. The square root of 90000 is expressed in both radical and exponential forms. In radical form, it is expressed as √90000, whereas (90000)^(1/2) in exponential form. √90000 = 300, which is a rational number because it can 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. Since 90000 is a perfect square, we can use the prime factorization method and also verify using the long division method. Let us now learn these methods: 

• Prime factorization method 
• Long division method

The product of prime factors is the prime factorization of a number. Now let us look at how 90000 is broken down into its prime factors.

Step 1: Finding the prime factors of 90000 Breaking it down, we get 2 × 2 × 3 × 3 × 5 × 5 × 5 × 5 = 2^2 × 3^2 × 5^4

Step 2: Now we found out the prime factors of 90000. The next step is to make pairs of those prime factors. Since 90000 is a perfect square, all the digits of the number can be grouped in pairs.

Therefore, calculating the square root using prime factorization is straightforward: √90000 = √(2^2 × 3^2 × 5^4) = 2 × 3 × 5^2 = 300.

The long division method is another way to find the square root of numbers, especially perfect squares. 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. For 90000, we group it as (90) (000).

Step 2: Now we need to find a number whose square is less than or equal to the first group, 90. We can choose 9, as 9 × 9 = 81. Now the quotient is 9, and the remainder is 90 - 81 = 9.

Step 3: Bring down the next pair of zeros to the remainder, making it 900.

Step 4: Double the quotient we obtained, which is 9, to get 18.

Step 5: Find a digit x such that 18x × x is less than or equal to 900. We find that x = 5, as 185 × 5 = 925, which is greater than 900, so we choose x = 4, as 184 × 4 = 736.

Step 6: Subtract 736 from 900, leaving a remainder of 164.

Step 7: Bring down the next pair of zeros to make it 16400.

Step 8: Now, double the new quotient formed, 94, to get 188.

Step 9: Find a digit x such that 188x × x is less than or equal to 16400. x = 8 works, as 1888 × 8 = 15104.

Step 10: Subtract 15104 from 16400, leaving a remainder of 1296.

Step 11: Continue this process until you reach a remainder of 0 or get the desired precision.

The quotient will be 300, confirming that √90000 = 300.

The approximation method involves finding the nearest perfect squares around 90000 and estimating the square root accordingly. However, since 90000 is a perfect square, we already know the exact square root is 300.

• 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.
   
• Rational number: A rational number is a number that can be written in the form of p/q, where p and q are integers and q is not equal to zero.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 90000 is a perfect square as it is 300 × 300.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: The prime factorization of 90000 is 2^2 × 3^2 × 5^4.
   
• Long division method: The long division method is a technique used to find the square root of numbers, especially when it involves decimals or non-perfect squares.