Square Root of 416

The square root is the inverse of the square of the number. 416 is not a perfect square. The square root of 416 is expressed in both radical and exponential form. In the radical form, it is expressed as √416, whereas (416)^(1/2) in the exponential form. √416 ≈ 20.396, 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, for non-perfect square numbers, 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 416 is broken down into its prime factors.

Step 1: Finding the prime factors of 416 Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 13: 2^5 × 13

Step 2: Now that we have found the prime factors of 416, the next step is to make pairs of those prime factors. Since 416 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating the square root of 416 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 numbers around 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 416, we need to group it as 16 and 4.

Step 2: Now we need to find n whose square is less than or equal to 4. We can say n as ‘2’ because 2 × 2 = 4. Now the quotient is 2, and after subtracting 4 from 4, the remainder is 0.

Step 3: Now let us bring down 16, which is the new dividend. Add the old divisor with the same number, 2 + 2 = 4, which will be our new divisor.

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

Step 5: The next step is finding 4n × n ≤ 16. Let us consider n as 4, now 4 × 4 = 16.

Step 6: Subtract 16 from 16. The difference is 0, and the quotient is 20.

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 0.

Step 8: Now we need to find the new divisor that is 403 because 403 × 0 = 0.

Step 9: The quotient is 20.3

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

So the square root of √416 ≈ 20.396

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 416 using the approximation method.

Step 1: Now we have to find the closest perfect square of √416. The smallest perfect square less than 416 is 400 and the largest perfect square greater than 416 is 441. √416 falls somewhere between 20 and 21.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Using the formula (416 - 400) ÷ (441 - 400) = 16 ÷ 41 ≈ 0.390 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 20 + 0.390 ≈ 20.390, so the square root of 416 is approximately 20.396.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 4^2 = 16, and the square root of 16 is √16 = 4.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction, where both the numerator and denominator are integers, and the denominator is not zero.

• Principal square root: A number has both positive and negative square roots, but we usually refer to the positive square root as the principal square root due to its practical applications.

• Prime factorization: Prime factorization is the process of breaking down a number into its prime factors.

• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point, such as 20.396.