Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields such as vehicle design, finance, etc. Here, we will discuss the 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:
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.
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let us look at a few of these mistakes in detail.
Can you help Max find the area of a square box if its side length is given as √473?
The area of the square is approximately 223.36 square units.
The area of the square = side^2.
The side length is given as √473.
Area of the square = side^2 = √473 x √473 ≈ 21.73 x 21.73 ≈ 472.78.
Therefore, the area of the square box is approximately 472.78 square units.
A square-shaped building measuring 473 square feet is built; if each of the sides is √473, what will be the square feet of half of the building?
236.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 473 by 2 = we get 236.5.
So half of the building measures 236.5 square feet.
Calculate √473 x 5.
Approximately 108.63
The first step is to find the square root of 473, which is approximately 21.73.
The second step is to multiply 21.73 with 5.
So, 21.73 x 5 ≈ 108.65.
What will be the square root of (441 + 32)?
The square root is approximately 21.73.
To find the square root, we need to find the sum of (441 + 32). 441 + 32 = 473, and √473 ≈ 21.73.
Therefore, the square root of (441 + 32) is approximately ±21.73.
Find the perimeter of the rectangle if its length ‘l’ is √473 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 119.46 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√473 + 38) ≈ 2 × (21.73 + 38) ≈ 2 × 59.73 ≈ 119.46 units.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.