Square Root of 243

The square root is the inverse of the square of the number. 243 is not a perfect square. The square root of 243 is expressed in both radical and exponential form. In the radical form, it is expressed as √243, whereas (243)^(1/2) in the exponential form. √243 ≈ 15.588, 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 applicable, and methods like 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 243 is broken down into its prime factors:

Step 1: Finding the prime factors of 243 Breaking it down, we get 3 x 3 x 3 x 3 x 3: 3^5

Step 2: Now we found out the prime factors of 243. The next step is to make pairs of those prime factors. Since 243 is not a perfect square, the digits of the number can’t be grouped into pairs completely. Therefore, calculating 243 using prime factorization directly is not feasible for finding an exact integer 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 digits from right to left. In the case of 243, we group it as 43 and 2.

Step 2: Identify n whose square is less than or equal to 2. We take n as 1 because 1 x 1 ≤ 2. Now, the quotient is 1, and the remainder is 1 after subtracting 1 from 2.

Step 3: Bring down 43, making the new dividend 143. Double the divisor (1) to get 2.

Step 4: Find the greatest digit (n) such that 2n x n ≤ 143. We choose n = 5, as 25 x 5 = 125.

Step 5: Subtract 125 from 143, leaving a remainder of 18, and update the quotient to 15.

Step 6: Since the remainder is less than the divisor, add a decimal point and bring down two zeroes, making the new dividend 1800.

Step 7: Find the new divisor, 310, and iterate the process to yield more decimal places, ultimately finding the square root to a desired precision.

So the square root of √243 is approximately 15.588.

The approximation method is another way to find square roots, and 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 243 using the approximation method.

Step 1: Identify the closest perfect squares around 243. The nearest perfect squares are 225 (15^2) and 256 (16^2). Thus, √243 falls between 15 and 16.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Using the formula: (243 - 225) ÷ (256 - 225) ≈ 0.58

Step 3: Add the decimal value to the smaller integer: 15 + 0.58 = 15.58 Therefore, the square root of 243 is approximately 15.58.

• Square root: A square root is the inverse operation of squaring a number. For example, if 5^2 = 25, then the square root of 25 is 5.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction, meaning it cannot be written as p/q, where p and q are integers, and q ≠ 0.
   
• Principal square root: The principal square root of a number is its non-negative square root, which is usually the one considered in practical applications.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 243 is 3^5.
   
• Decimal: A decimal is a number that contains a whole number and a fractional part separated by a decimal point, such as 7.86, 8.65, and 9.42.