Square Root of 443

The square root is the inverse of the square of the number. 443 is not a perfect square. The square root of 443 is expressed in both radical and exponential form. In the radical form, it is expressed as √443, whereas (443)^(1/2) in the exponential form. √443 ≈ 21.045, 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 443 is broken down into its prime factors:

Step 1: Finding the prime factors of 443 Breaking it down, we get 443 = 443^1, as 443 is a prime number itself.

Step 2: Since 443 is not a perfect square, calculating its square root using prime factorization is not straightforward.

Hence, we use other methods like long division or approximation for non-perfect squares.

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 443, we group it as 43 and 4.

Step 2: Now we need to find a number whose square is less than or equal to 4. We can say n is '2' because 2 x 2 = 4. Now the quotient is 2 after subtracting 4 - 4, the remainder is 0.

Step 3: Now let us bring down 43, which is the new dividend. Add the old divisor with the same number 2 + 2, we get 4, which will be our new divisor.

Step 4: Now we get 4n as the new divisor, we need to find the value of n such that 4n x n ≤ 43. It turns out 4 x 5 x 5 = 100 is too large, so let's try smaller.

Step 5: The next step is finding 4n x n ≤ 43, let us consider n as 1, now 4 x 1 x 1 = 4.

Step 6: Subtract 43 from 4, the difference is 39, and the quotient is 21.

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. Now the new dividend is 3900.

Step 8: Now we need to find the new divisor. Let's try higher n values to get closer to 3900, e.g., 421 x 9 = 3789.

Step 9: Subtracting 3789 from 3900 gives the result 111.

Step 10: Now the quotient is 21.0, and we continue until we achieve the desired precision.

So the square root of √443 is approximately 21.045.

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 443 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √443. The smallest perfect square less than 443 is 400, and the largest perfect square greater than 443 is 441. √443 falls somewhere between 21 and 22.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula: (443 - 400) ÷ (441 - 400) = 43 ÷ 41 ≈ 1.048 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 + 0.048 = 21.048, so the square root of 443 is approximately 21.048.

• Square root: A square root is the inverse of a square. For example: 4^2 = 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 has only two factors, 1 and itself. Example: 443 is a prime number.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 144 is a perfect square because 12 x 12 = 144.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.