Square Root of 463

The square root is the inverse of the square of the number. 463 is not a perfect square. The square root of 463 is expressed in both radical and exponential form. In the radical form, it is expressed as √463, whereas (463)^(1/2) in the exponential form. √463 ≈ 21.5, 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

• 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 463 is broken down into its prime factors.

Step 1: Finding the prime factors of 463. 463 is a prime number itself, so it cannot be broken down further. Since 463 is not a perfect square, calculating it using prime factorization is not feasible.

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 463, we take 63 and 4 as groups.

Step 2: Now we need to find n whose square is less than or equal to 4. We can take n as '2' because 2 × 2 = 4. Now the quotient is 2, and after subtracting 4-4, the remainder is 0.

Step 3: Now bring down 63, which is the new dividend. Add the old divisor with the same number, 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n, and we need to find n such that 4n × n ≤ 63. If we consider n as 1, then 41 × 1 = 41.

Step 5: Subtract 63 from 41; the difference is 22, and the quotient is 21.

Step 6: 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. Now the new dividend is 2200.

Step 7: The new divisor is 422. We find n = 5 because 425 × 5 = 2125.

Step 8: Subtracting 2125 from 2200 gives a result of 75.

Step 9: Now the quotient is 21.5.

Step 10: Continue doing these steps until we get two numbers after the decimal point. If needed, continue until the remainder is zero.

So the square root of √463 ≈ 21.5

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 463 using the approximation method.

Step 1: Find the closest perfect square of √463. The smallest perfect square less than 463 is 441 (21²), and the largest perfect square greater than 463 is 484 (22²). √463 falls somewhere between 21 and 22.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (463 - 441) ÷ (484 - 441) = 22/43 ≈ 0.51.

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 21 + 0.51 = 21.51.

Therefore, the square root of 463 is approximately 21.51.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root; that is, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Prime number: A prime number is a number greater than 1 that has no divisors other than 1 and itself.
   
• Long division method: A technique used to find the square root of non-perfect squares by repeatedly dividing and averaging.
   
• Decimal approximation: A way to express an irrational number as a decimal that is not exact, often used for practical calculations.