Square Root of 80000

The square root is the inverse of the square of the number. 80000 is not a perfect square. The square root of 80000 is expressed in both radical and exponential form. In the radical form, it is expressed as √80000, whereas (80000)^(1/2) in the exponential form. √80000 = 282.8427125, 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 80000 is broken down into its prime factors.

Step 1: Finding the prime factors of 80000 Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 × 5: 2^5 × 5^5

Step 2: Now we found out the prime factors of 80000. The second step is to make pairs of those prime factors. Since 80000 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely.

Therefore, calculating 80000 using prime factorization is not feasible to get an exact integer square root.

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 80000, we group it as 800 and 00.

Step 2: Now we need to find n whose square is 8. We can say n as ‘2’ because 2 × 2 is lesser than or equal to 8. Now the quotient is 2; after subtracting 4 from 8, the remainder is 4.

Step 3: Now let us bring down 000, making the new dividend 400. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor is 40, and we need to find n such that 40n × n ≤ 400. Let n be 9; now 409 × 9 = 3681.

Step 5: Subtract 3681 from 40000, the difference is 319, and the quotient is 280.

Step 6: 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 31900.

Step 7: Find the new divisor, which is 282 because 2820 × 1 = 2820.

Step 8: Subtracting 2820 from 31900 gives a result of 369. Step 9: The quotient is now 282.8

Step 10: Continue doing these steps until we get sufficient numbers after the decimal point. Suppose if there is no decimal value continue till the remainder is zero.

So the square root of √80000 is approximately 282.84.

The approximation method is another method for finding the 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 80000 using the approximation method.

Step 1: Now we have to find the closest perfect square of √80000. The smallest perfect square less than 80000 is 64000 (which is 80^2) and the largest perfect square greater than 80000 is 81000 (which is 90^2). √80000 falls somewhere between 282 and 283.

Step 2: Now we need to apply the linear approximation formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Using the formula (80000 - 64000) ÷ (81000 - 64000) = 0.941176 Using the formula, we identified the decimal point of our square root. The next step is adding the value 282 (closest integer root) to the decimal number: 282 + 0.94 = 282.94.

So the square root of 80000 is approximately 282.94.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse of squaring is taking 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.
   
• Decimal: A decimal is a number that contains a whole number and a fractional part separated by a decimal point, for example: 7.86, 8.65, and 9.42 are decimals.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that is used in most real-world applications. This positive root is called the principal square root.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 60 is 2^2 × 3 × 5.