Square Root of 33.3

The square root is the inverse of the square of a number. 33.3 is not a perfect square. The square root of 33.3 is expressed in both radical and exponential form. In the radical form, it is expressed as √33.3, whereas (33.3)¹/² in the exponential form. √33.3 ≈ 5.7717, 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 prime factorization method is not applied. Instead, methods such as the long-division method and the approximation method are used. Let us now learn these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. However, since 33.3 is not a whole number, traditional prime factorization is not applicable directly. For decimal numbers, approximation methods are more suitable.

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, consider the integer part of 33.3, which is 33. Pair 33 from right to left.

Step 2: Find a number whose square is less than or equal to 33. The number is 5, since 5 × 5 = 25.

Step 3: Subtract 25 from 33, we get a remainder of 8. Bring down two zeroes to make it 800.

Step 4: Double the quotient (5) and write it beside the divisor as 10. We need to find a digit x such that 10x × x is less than or equal to 800.

Step 5: By trial, 107 × 7 = 749.

Step 6: Subtract 749 from 800, we get 51. Bring down two more zeroes to make it 5100.

Step 7: Double the current quotient to get 114 and continue the process.

Step 8: Continue this process until the desired decimal places are reached.

So the square root of √33.3 ≈ 5.7717

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 33.3 using the approximation method.

Step 1: Now we have to find the closest perfect square of √33.3.

The closest perfect squares are 25 (5²) and 36 (6²).

√33.3 falls between 5 and 6.

Step 2: Now we apply the formula:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square)

Using the formula, (33.3 - 25) / (36 - 25) = 8.3 / 11 ≈ 0.7545

Adding this to 5: 5 + 0.7545 = 5.7545, so the square root of 33.3 ≈ 5.7717

• Square root: A square root is the inverse operation of squaring a number. For example, if 4² = 16, then √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction. It has a non-repeating, non-terminating decimal expansion.
   
• Principal square root: The principal square root refers to the positive square root of a number. For example, the principal square root of 16 is 4.
   
• Decimal: A decimal is a number expressed in the base-10 numeral system, which includes a whole number and a fractional part separated by a decimal point. For example, 33.3 is a decimal.
   
• Long division method: A procedure used to find the square root of non-perfect squares by dividing and averaging to reach an approximation.