Square Root of 1616

The square root is the inverse of the square of a number. 1616 is not a perfect square. The square root of 1616 is expressed in both radical and exponential form. In the radical form, it is expressed as √1616, whereas (1616)^(1/2) in the exponential form. √1616 ≈ 40.1995, 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 like 1616, methods such as the long-division method and approximation method are used to find the square root. 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 explore how 1616 is broken down into its prime factors.

Step 1: Finding the prime factors of 1616 Breaking it down, we get 2 x 2 x 2 x 2 x 101: 2^4 x 101

Step 2: Now that we have found the prime factors of 1616, the next step is to make pairs of those prime factors. Since 1616 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating 1616 using prime factorization is not feasible for finding an exact 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: Begin by grouping the numbers from right to left. In the case of 1616, group it as 16 and 16.

Step 2: Find n whose square is less than or equal to 16. Here, n is 4 because 4^2 = 16. Now the quotient is 4 after subtracting 16-16, the remainder is 0.

Step 3: Bring down the next pair of digits, which is 16. Add the old divisor with the same number 4 + 4 to get 8, which will be our new divisor.

Step 4: The new divisor will be 8n, where we need to find the value of n.

Step 5: Find 8n × n ≤ 16. Let us consider n as 2, now 8 x 2 x 2 = 32

Step 6: Subtract 16 from 32; since 32 is greater, choose n such that the multiplication is less than or equal to the dividend. The closest possible n is 0.

Step 7: Since the dividend is less than the divisor, add a decimal point. This allows us to add two zeroes to the dividend. Now the new dividend is 1600.

Step 8: Find the new divisor, which is 80n, where n is 2 because 802 ✖ 2 = 1600.

Step 9: Subtract 1600 from 1600; the result is 0.

Step 10: Now the quotient is 40.2

Step 11: Continue these steps until there are two decimal places in the quotient or the remainder is zero.

So the square root of √1616 is approximately 40.20.

The approximation method is another simple way to find the square roots. Let's learn how to find the square root of 1616 using this method.

Step 1: Identify the closest perfect squares to √1616.

The smallest perfect square less than 1616 is 1600, and the largest perfect square greater than 1616 is 1681.

√1616 falls somewhere between 40 and 41.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula, (1616 - 1600) ÷ (1681 - 1600) = 16 ÷ 81 ≈ 0.198.

Adding this to the smaller integer root, 40 + 0.198 = 40.198, so the square root of 1616 is approximately 40.198.

• Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 16, and the inverse of the square is the square root, so √16 = 4.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where p and q are integers and q ≠ 0.
   
• Principal square root: A number has both positive and negative square roots, but the positive square root is commonly used in practical applications, hence known as the principal square root.
   
• Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 1616 is 2^4 × 101.
   
• Decimal: A number with a whole number and a fractional part separated by a decimal point, e.g., 7.86, 8.65, and 9.42.