Square Root of 4046

The square root is the inverse of the square of a number. 4046 is not a perfect square. The square root of 4046 is expressed in both radical and exponential form. In the radical form, it is expressed as √4046, whereas in exponential form it is (4046)^(1/2). √4046 ≈ 63.593, 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 4046 is broken down into its prime factors:

Step 1: Finding the prime factors of 4046 Breaking it down, we get 2 x 2023. 2023 is further factorized as 7 x 17 x 17.

Step 2: Now we found out the prime factors of 4046. The second step is to make pairs of those prime factors. Since 4046 is not a perfect square, the digits of the number can’t be perfectly grouped in pairs.

Therefore, calculating 4046 using prime factorization does not yield an exact value.

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 4046, we need to group it as 46 and 40.

Step 2: Now we need to find n whose square is 40. We can say n as ‘6’ because 6 x 6 = 36 is lesser than or equal to 40. Now the quotient is 6; after subtracting 36 from 40, the remainder is 4.

Step 3: Now let us bring down 46, which is the new dividend. Add the old divisor with the same number 6 + 6 we get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 446. Let us consider n as 3, now 12 x 3 x 3 = 369.

Step 6: Subtract 369 from 446, the difference is 77, and the quotient is 63.

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 7700.

Step 8: Now we need to find the new divisor that is 635 because 635 x 5 = 3175.

Step 9: Subtracting 3175 from 7700, we get the result 4525.

Step 10: Now the quotient is 63.5.

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

So the square root of √4046 is approximately 63.59.

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 4046 using the approximation method.

Step 1: Now we have to find the closest perfect square of √4046. The smallest perfect square less than 4046 is 3969 (63^2), and the largest perfect square greater than 4046 is 4096 (64^2). √4046 falls somewhere between 63 and 64.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (4046 - 3969) ÷ (4096 - 3969) = 77 ÷ 127 = 0.606. 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 63 + 0.606 = 63.606. So the square root of 4046 is approximately 63.606.

• Square root: A square root is the inverse of a square. For example, 8^2 = 64, and the inverse of the square is the square root, which is √64 = 8.
   
• 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 why it is also known as the principal square root.
   
• Prime factorization: Prime factorization involves breaking down a number into its smallest prime factors.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 64 is a perfect square because it is 8^2.