Square Root of 433

The square root is the inverse operation of squaring a number. The number 433 is not a perfect square. The square root of 433 can be expressed in both radical and exponential forms. In radical form, it is expressed as √433, whereas in exponential form it is (433)^(1/2). √433 ≈ 20.809, which is an irrational number because it cannot be expressed as a fraction of two integers.

The prime factorization method is typically used for perfect square numbers. However, for non-perfect squares like 433, we use the long division method and the approximation method. Let us now learn the following methods:

• Long division method

• Approximation method

The long division method is particularly useful for finding the square roots of non-perfect squares. Here are the steps to find the square root using this method:

Step 1: Group the digits of 433 from right to left. Since 433 has only three digits, we consider it as 4 | 33.

Step 2: Find the largest number whose square is less than or equal to 4. This number is 2, because 2^2 = 4. Place 2 as the first digit of the quotient. Subtract 4 from 4 to get a remainder of 0.

Step 3: Bring down 33 to make it the new dividend. Double the divisor (2), which is now 4.

Step 4: Find a digit 'n' such that 4n × n is less than or equal to 33. In this case, n=8, because 48 × 8 = 384, which is greater than 33. Thus, n=7, because 47 × 7 = 329, which is closer.

Step 5: Subtract 329 from 330 (33 with a decimal point added) to get 1. Let the quotient be 20.7.

Step 6: Continue the process to find more decimals. Add two zeros to the remainder to get 100. Repeat the process to get a more precise result.

So, the approximate square root of √433 is 20.809.

The approximation method is another way to find square roots, which is simpler for a quick estimate. Here is how to approximate the square root of 433:

Step 1: Identify the closest perfect squares around 433. The perfect squares are 400 (20^2) and 441 (21^2). Therefore, √433 lies between 20 and 21.

Step 2: Use interpolation to approximate: (433 - 400) / (441 - 400) = 33 / 41 ≈ 0.805 Adding this to the lower bound gives 20 + 0.805 = 20.805. Thus, √433 ≈ 20.809.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4.

• Irrational number: An irrational number cannot be expressed as a fraction of two integers, where the denominator is not zero. Example: √2.

• Approximation: The process of finding a value that is close to, but not exactly equal to, a certain number. Example: π ≈ 3.14.

• Long division method: A method used to find the square root of a number by dividing and averaging in a step-by-step manner.

• Prime number: A number greater than 1 that has no positive divisors other than 1 and itself. Example: 433.