Square Root of 8000

The square root is the inverse of the square of the number. 8000 is not a perfect square. The square root of 8000 is expressed in both radical and exponential form. In the radical form, it is expressed as √8000, whereas (8000)^(1/2) in the exponential form. √8000 = 89.4427, 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 8000 is broken down into its prime factors.

Step 1: Finding the prime factors of 8000 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5: 2^5 x 5^3

Step 2: Now that we have found the prime factors of 8000, the second step is to make pairs of those prime factors. Since 8000 is not a perfect square, the digits of the number can’t be grouped in pairs completely. Therefore, calculating 8000 using prime factorization requires further calculation.

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 8000, we need to group it as 80 and 00.

Step 2: Now we need to find n whose square is 64. We can say n as ‘8’ because 8 x 8 = 64, which is less than or equal to 80. Now the quotient is 8 after subtracting 80 - 64; the remainder is 16.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 8 + 8, we get 16, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 16n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 16n × n ≤ 1600. Let us consider n as 9, now 16 x 9 = 144

Step 6: Subtract 1600 from 144, the difference is 1600 - 144 = 1456, and the quotient is 89.

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 145600.

Step 8: Now we need to find the new divisor that is 178 because 178 x 8 = 1424

Step 9: Subtracting 1424 from 1456, we get the result 32.

Step 10: Now the quotient is 89.4

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 √8000 is approximately 89.44.

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 8000 using the approximation method.

Step 1: Now we have to find the closest perfect square of √8000. The smallest perfect square less than 8000 is 7921, and the largest perfect square more than 8000 is 8100. √8000 falls somewhere between 89 and 90.

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 (8000 - 7921) ÷ (8100 - 7921) = 0.4427 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 89 + 0.4427 = 89.4427, so the square root of 8000 is approximately 89.4427.

• 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 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 process of expressing a number as the product of its prime factors.
   
• Long division method: A step-by-step method used to find the square root of non-perfect squares, involving division and subtraction to approximate the root.