Square Root of 18000

The square root is the inverse of the square of the number. 18000 is not a perfect square. The square root of 18000 is expressed in both radical and exponential form. In the radical form, it is expressed as √18000, whereas (18000)^(1/2) in the exponential form. √18000 ≈ 134.16408, 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 long-division and approximation methods 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 18000 is broken down into its prime factors.

Step 1: Finding the prime factors of 18000 Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 5: 2³ x 3² x 5⁴

Step 2: Now we found out the prime factors of 18000. The next step is to make pairs of those prime factors. Since 18000 is not a perfect square, therefore the digits of the number can’t be grouped in a complete pair.

Therefore, calculating 18000 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 18000, we need to group it as 00, 80, and 18.

Step 2: Now we need to find n whose square is less than or equal to 18. We can say n is ‘4’ because 4 x 4 = 16, which is less than 18. Now the quotient is 4 after subtracting 16 from 18, the remainder is 2.

Step 3: Now let us bring down 80 which is the new dividend. Add the old divisor with the same number 4 + 4 we get 8, making it our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 280. Let us consider n as 3, now 83 x 3 = 249.

Step 6: Subtract 249 from 280, the difference is 31, and the quotient is 43.

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 zeros to the dividend. Now the new dividend is 3100.

Step 8: Now we need to find the new divisor that is 866 because 866 x 3 = 2598

Step 9: Subtracting 2598 from 3100, we get the result 502.

Step 10: Now the quotient is 134.1

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values continue till the remainder is zero.

So the square root of √18000 is approximately 134.16

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 18000 using the approximation method.

Step 1: Now we have to find the closest perfect square of √18000. The smallest perfect square less than 18000 is 17689, and the largest perfect square greater than 18000 is 18225. √18000 falls somewhere between 133 and 135.

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 (18000 - 17689) ÷ (18225-17689) ≈ 0.16408. 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 134 + 0.16408 ≈ 134.16408, so the square root of 18000 is approximately 134.16408.

• 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.

• Principal square root: A number has both positive and negative square roots; however, it is usually the positive square root that is used in real-world applications. This is known as the principal square root.

• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: The prime factorization of 18000 is 2³ x 3² x 5⁴.

• Decimal: A decimal is a way of representing numbers that have a whole number part and a fractional part, separated by a decimal point. Example: 134.16 is a decimal.