Square Root of 545

The square root is the inverse of the square of a number. 545 is not a perfect square. The square root of 545 is expressed in both radical and exponential form. In the radical form, it is expressed as √545, whereas (545)^(1/2) in the exponential form. √545 ≈ 23.3452, 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 the long-division method and approximation method are used. Let us now learn the following methods:

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

The product of prime factors is the prime factorization of a number. Now let us look at how 545 is broken down into its prime factors.

Step 1: Finding the prime factors of 545 Breaking it down, we get 5 × 109.

Step 2: Now we have found the prime factors of 545. Since 545 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 545 using prime factorization as a perfect square is not possible.

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, we need to group the numbers from right to left. In the case of 545, we group it as 45 and 5.

Step 2: Now we need to find n whose square is less than or equal to 5. We can say n is '2' because 2 × 2 = 4, which is less than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 45, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor prefix.

Step 4: The new divisor will be the sum of the previous quotient and the new digit. Now we get 4n as the new divisor, and we need to find the value of n such that 4n × n ≤ 145.

Step 5: Considering n as 3, 43 × 3 = 129.

Step 6: Subtract 129 from 145, the difference is 16, and the quotient now is 23.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 1600.

Step 8: Now we need to find the new divisor, which is 466 because 466 × 3 = 1398.

Step 9: Subtracting 1398 from 1600, we get the result 202. Step 10: Now the quotient is 23.3.

Step 11: Continue these steps until we get the desired accuracy.

So the square root of √545 is approximately 23.345.

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 545 using the approximation method.

Step 1: Now we have to find the closest perfect square of √545. The smallest perfect square less than 545 is 529, and the largest perfect square greater than 545 is 576. √545 falls somewhere between 23 and 24.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula, (545 - 529) / (576 - 529) = 16 / 47 ≈ 0.34. Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number, which is 23 + 0.34 = 23.34.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16, and the inverse of the square is the square root, so √16 = 4.

• Irrational number: An irrational number cannot be written as a simple fraction, for instance, the decimal for √545 goes on forever without repeating.

• Perfect square: A perfect square is an integer that is the square of an integer. Examples include 1, 4, 9, 16, 25, etc.

• Radical form: The expression of a number as a square root is its radical form, such as √545.

• Long division method: A method used to find the square root of a number by dividing it into manageable steps.