Square Root of 487

The square root is the inverse of squaring a number. 487 is not a perfect square. The square root of 487 is expressed in both radical and exponential form. In radical form, it is expressed as √487, whereas in exponential form, it is expressed as (487)^(1/2). √487 ≈ 22.068, 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 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 487 is broken down into its prime factors:

Step 1: Finding the prime factors of 487. 487 is a prime number, so it cannot be broken down further into other prime factors. Thus, it cannot be simplified using the prime factorization method. Calculating the square root of 487 using prime factorization is not feasible.

The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step:

Step 1: Start by grouping the digits from right to left in pairs. In the case of 487, we have 4 and 87.

Step 2: Find a number n whose square is less than or equal to 4. This number is 2, because 2 × 2 = 4.

Step 3: Subtract 4 from 4, which leaves a remainder of 0, and bring down the next pair, 87.

Step 4: Double the divisor (2) to get 4, and find a new digit (n) such that 4n × n ≤ 87. This digit is 2, because 42 × 2 = 84.

Step 5: Subtract 84 from 87, resulting in a remainder of 3.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down a pair of zeros to the remainder to make it 300.

Step 7: The new divisor is 44, and find n such that 44n × n ≤ 300. The number is 6, because 446 × 6 = 2676.

Step 8: Subtract 2676 from 3000 to get a remainder of 324.

Step 9: Continue the process to get the desired precision.

So the square root of √487 ≈ 22.068.

The approximation method is another method for finding square roots. Now let us learn how to find the square root of 487 using the approximation method.

Step 1: Determine the closest perfect squares surrounding √487. The closest perfect squares are 484 (22^2) and 529 (23^2). Thus, √487 lies between 22 and 23.

Step 2: Apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Applying the formula: (487 - 484) ÷ (529 - 484) = 3 ÷ 45 ≈ 0.067. Adding this to 22 gives us 22 + 0.067 ≈ 22.067.

Hence, the square root of 487 is approximately 22.067.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse is the square root, √16 = 4.
   
• Irrational number: A number that cannot be expressed as a simple fraction (p/q, where p and q are integers and q ≠ 0), such as √487.
   
• Prime number: A number greater than 1 with no divisors other than 1 and itself, such as 487.
   
• Decimal: A number that includes a fractional part, such as 22.068.
   
• Long division method: A technique for finding the square root of non-perfect squares by dividing the number into smaller, more manageable parts.