Square Root of 6409

The square root is the inverse of the square of the number. 6409 is not a perfect square. The square root of 6409 is expressed in both radical and exponential form. In the radical form, it is expressed as √6409, whereas (6409)^(1/2) in the exponential form. √6409 ≈ 80.0587, 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 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 6409 is broken down into its prime factors.

Step 1: Finding the prime factors of 6409 Breaking it down, we find that 6409 is not easily broken down into smaller prime factors.

Step 2: Since 6409 is not a perfect square, calculating 6409 using the prime factorization method 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 6409, we need to group it as 64 and 09.

Step 2: Now we need to find n whose square is 64. We can say n as ‘8’ because 8 × 8 = 64. Now the quotient is 8, and after subtracting 64 from 64, the remainder is 0.

Step 3: Now let us bring down 09, which is the new dividend. Double the old divisor 8, which gives us 16 as our new divisor.

Step 4: Find n such that 16n × n ≤ 09. We can say n = 0, because 160 × 0 = 0.

Step 5: The quotient is 80.

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 900.

Step 7: Now we need to find the new divisor, 160, which gives us 1600 × 0 ≤ 900.

Step 8: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √6409 is approximately 80.06.

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 6409 using the approximation method.

Step 1: Now we have to find the closest perfect square of √6409.

The smallest perfect square less than 6409 is 6400, and the largest perfect square greater than 6409 is 6561. √6409 falls somewhere between 80 and 81.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (6409 - 6400) ÷ (6561 - 6400) ≈ 0.0587.

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 80 + 0.0587 ≈ 80.06.

So the square root of 6409 is approximately 80.06.

• 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 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 the principal square root.

• Long division method: A method used to find the square roots of non-perfect squares using division steps to approximate the root.

• Approximation method: A method used to estimate the square root of a non-perfect square by finding the closest perfect squares.