Square Root of 543

The square root is the inverse of squaring a number. 543 is not a perfect square. The square root of 543 can be expressed in both radical and exponential forms. In radical form, it is expressed as √543, whereas in exponential form as (543)^(1/2). √543 ≈ 23.310, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is typically used for perfect square numbers. However, for non-perfect square numbers like 543, the long-division method and approximation method are used. Let us explore these methods:

1. Prime factorization method
2. Long division method
3. Approximation method

Prime factorization involves breaking down a number into its prime factors. Let's see how 543 is factored:

Step 1: Find the prime factors of 543. Breaking it down, we get 3 x 181: 3¹ x 181¹

Step 2: As 543 is not a perfect square, we cannot group the prime factors into pairs.

Therefore, calculating √543 using prime factorization is not feasible.

The long division method is effective for non-perfect squares. Let's find the square root of 543 step by step:

Step 1: Group the digits of 543 from right to left. In this case, 543 is grouped as 5 and 43.

Step 2: Find a number whose square is less than or equal to 5. This number is 2 because 2 x 2 = 4. Subtracting gives a remainder of 1.

Step 3: Bring down the next pair, 43, making the new dividend 143. Double the previous quotient (2) and write it as the new divisor’s leading digit, making it 4_.

Step 4: Find the largest digit for the blank space (n) such that 4n x n ≤ 143. Here, n is 3, as 43 x 3 = 129.

Step 5: Subtract 129 from 143 to get 14. The new quotient is 23.

Step 6: Since 14 is less than 40, we append ".00" to the dividend to continue the process with 1400.

Step 7: Continue this method until the desired decimal places are found.

The square root of 543 is approximately 23.310.

The approximation method is a simpler way to find square roots. Here's how to approximate the square root of 543:

Step 1: Identify the closest perfect squares around 543. The nearest are 529 (23²) and 576 (24²). Therefore, √543 lies between 23 and 24.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).

Using the formula: (543 - 529) / (576 - 529) = 14 / 47 ≈ 0.298. Add this decimal to the smaller number: 23 + 0.298 = 23.298, so the square root of 543 is approximately 23.298.

• Square root: A square root is the inverse operation of squaring a number. For example, 5² = 25, and the square root of 25 is √25 = 5.

• Irrational number: An irrational number cannot be expressed as a simple fraction. It is a non-repeating, non-terminating decimal, such as √543.

• Principal square root: This is the non-negative square root of a number. For example, the principal square root of 25 is 5.

• Approximation: This refers to finding a value close to the exact answer, often used when working with irrational numbers.

• Long Division: A step-by-step division process that can be used to find square roots for non-perfect squares.