Square Root of 473

The square root is the inverse of the square of the number. 473 is not a perfect square. The square root of 473 is expressed in both radical and exponential form. In radical form, it is expressed as √473, whereas (473)^(1/2) in exponential form. √473 ≈ 21.72556, which is an irrational number because it cannot be expressed in the form 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 the 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 473 is broken down into its prime factors.

Step 1: Finding the prime factors of 473

Breaking it down, we find that 473 is a product of prime numbers: 11 x 43

Step 2: Now we found out the prime factors of 473. The second step is to make pairs of those prime factors. Since 473 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 473 using prime factorization is not straightforward.

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 473, we group it as 73 and 4.

Step 2: Now we need to find n 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, and after subtracting 4 - 4, the remainder is 0.

Step 3: Now let us bring down 73, 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: The new divisor will be 4n. We need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 73. Let us consider n as 1, now 41 x 1 = 41.

Step 6: Subtract 73 from 41; the difference is 32, and the quotient is now 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 zeros to the dividend. Now the new dividend is 3200.

Step 8: Now we need to find the new divisor. Let n be 7 because 437 x 7 = 3059.

Step 9: Subtracting 3059 from 3200, we get the result 141.

Step 10: Now the quotient is 21.7

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there is no decimal value; continue until the remainder is zero.

So the square root of √473 is approximately 21.72.

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 473 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √473.

The smallest perfect square less than 473 is 441 (21^2), and the largest perfect square greater than 473 is 484 (22^2). √473 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 (473 - 441) ÷ (484 - 441) = 32 ÷ 43 ≈ 0.744

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 + 0.744 = 21.744.

So the square root of 473 is approximately 21.744.

• Square root: A square root is the inverse of a square. 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.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that is used in practical applications, known as the principal square root.
   
• Integers: The combination of whole numbers and negative numbers are integers, for example: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6,7…… are integers.
   
• 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.