Square Root of 16.5

The square root is the inverse of the square of the number. 16.5 is not a perfect square. The square root of 16.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √16.5, whereas (16.5)^(1/2) in the exponential form. √16.5 ≈ 4.062019, 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. Since 16.5 is not an integer, it cannot be directly factored into primes without expressing it in fractional form. Therefore, calculating 16.5 using prime factorization is not applicable.

The long division method is particularly used for non-perfect square numbers. In this method, we should find the closest perfect square numbers 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, consider 16.5 as 1650 by multiplying it by 100, which is equivalent to considering two decimal places.

Step 2: Now, group the digits in pairs from right to left: 16 and 50

Step 3: Find the largest number whose square is less than or equal to 16. That number is 4.

Step 4: Subtract 16 - 16 = 0, and bring down the next pair of digits, 50, making it 50.

Step 5: Double the quotient, 4, to get 8, which will be part of the new divisor.

Step 6: Find a digit, n, such that 8n × n ≤ 50. Here, 85 × 5 = 425, which fits the condition.

Step 7: Subtract 425 from 500, resulting in 75, and bring down another pair of zeros, making it 7500.

Step 8: Continue this process until the desired precision is achieved.

The square root of 16.5 is approximately 4.062.

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 16.5 using the approximation method.

Step 1: Find the closest perfect squares around 16.5. The closest perfect square less than 16.5 is 16, and the closest perfect square greater than 16.5 is 25.

Step 2: √16 = 4 and √25 = 5, so √16.5 falls between 4 and 5.

Step 3: Approximating further using linear interpolation: (16.5 - 16) / (25 - 16) = 0.5 / 9 ≈ 0.0556.

Step 4: Add this to the smaller square root: 4 + 0.0556 ≈ 4.0556.

Thus, the approximate square root of 16.5 is 4.056.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
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• Long division method: A step-by-step method used to find the square root of non-perfect square numbers by grouping and estimating digits.