Square Root of 9000

The square root is the inverse of the square of the number. 9000 is not a perfect square. The square root of 9000 is expressed in both radical and exponential form. In the radical form, it is expressed as √9000, whereas (9000)^(1/2) in the exponential form. √9000 ≈ 94.86833, 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, for non-perfect square numbers, the prime factorization method is not used; instead, long-division and approximation methods 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 9000 is broken down into its prime factors.

Step 1: Finding the prime factors of 9000

Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5: 2^3 x 3^2 x 5^3

Step 2: Now we found out the prime factors of 9000. The second step is to make pairs of those prime factors. Since 9000 is not a perfect square, therefore, the digits of the number can’t be grouped in perfect pairs. Therefore, calculating 9000 using prime factorization directly is not feasible.

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, we need to group the numbers from right to left. In the case of 9000, we need to group it as 90 and 00.

Step 2: Now we need to find n whose square is less than or equal to 90. We can say n is '9' because 9 x 9 = 81, which is less than 90. Now the quotient is 9, and after subtracting 81 from 90, the remainder is 9.

Step 3: Now let us bring down 00, making the new dividend 900. Add the old divisor with the same number: 9 + 9 = 18, which will be our new divisor.

Step 4: The new divisor will be 18, and we need to find the value of n.

Step 5: The next step is finding 18n × n ≤ 900. Let us consider n as 5; now 18 x 5 = 90, and 90 x 5 = 450, which is less than 900.

Step 6: Subtract 450 from 900; the difference is 450, and the quotient is 95.

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 45000.

Step 8: Now we need to find the new divisor, which is 189 because 1895 x 5 = 9475, which is less than 45000.

Step 9: Subtracting 9475 from 45000 gives a result of 35525.

Step 10: The quotient is 94.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √9000 is approximately 94.87.

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 9000 using the approximation method.

Step 1: Now we have to find the closest perfect square to √9000. The smallest perfect square less than 9000 is 8836 and the largest perfect square greater than 9000 is 9025. √9000 falls somewhere between 94 and 95.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula: (9000 - 8836) ÷ (9025 - 8836) = 0.8644

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 94 + 0.8644 = 94.8644, so the square root of 9000 is approximately 94.8644.

• 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, 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 the 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: The expression of a number as a product of prime numbers. For example, the prime factorization of 9000 is 2^3 x 3^2 x 5^3.
   
• 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.