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 and finance. Here, we will discuss the square root of 743.
The square root is the inverse of the square of the number. 743 is not a perfect square. The square root of 743 is expressed in both radical and exponential form. In the radical form, it is expressed as √743, whereas (743)^(1/2) in the exponential form. √743 ≈ 27.258, 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:
The product of prime factors is the prime factorization of a number. Now let us look at how 743 is broken down into its prime factors:
Step 1: Finding the prime factors of 743 743 is a prime number, so it cannot be broken down into smaller prime factors. Therefore, calculating 743 using prime factorization is impossible.
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 743, we need to group it as 43 and 7.
Step 2: Now we need to find n whose square is less than or equal to 7. We can say n is ‘2’ because 2 x 2 = 4, which is less than 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.
Step 3: Bring down 43 to make the new dividend 343. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, and we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 343. Let us consider n as 8, now 48 x 8 = 384 which is more than 343. So, let's try n as 7, 47 x 7 = 329.
Step 6: Subtract 329 from 343, the difference is 14, and the quotient becomes 27.
Step 7: Since the dividend is less than the new 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 1400.
Step 8: Now we need to find the new divisor which is 547 because 547 x 2 = 1094.
Step 9: Subtracting 1094 from 1400 we get the result 306.
Step 10: Now the quotient is 27.2
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue till the remainder is zero.
So the square root of √743 is approximately 27.26.
The approximation method is another method for finding square roots, and it is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 743 using the approximation method.
Step 1: Now we have to find the closest perfect square of √743. The smallest perfect square less than 743 is 729 (27^2) and the largest perfect square greater than 743 is 784 (28^2). √743 falls somewhere between 27 and 28.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (743 - 729) ÷ (784 - 729) = 0.2545. 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 27 + 0.2545 = 27.2545, so the square root of 743 is approximately 27.2545.
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division methods. Now let us look at a few of those mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √743?
The area of the square is approximately 551.29 square units.
The area of the square = side².
The side length is given as √743.
Area of the square = side² = √743 x √743 ≈ 27.258 x 27.258 ≈ 743.
Therefore, the area of the square box is approximately 551.29 square units.
A square-shaped garden measuring 743 square feet is built; if each of the sides is √743, what will be the square feet of half of the garden?
371.5 square feet
We can just divide the given area by 2 as the garden is square-shaped.
Dividing 743 by 2 gives us 371.5.
So half of the garden measures 371.5 square feet.
Calculate √743 x 3.
Approximately 81.774
The first step is to find the square root of 743, which is approximately 27.258. The second step is to multiply 27.258 by 3. So 27.258 x 3 ≈ 81.774.
What will be the square root of (729 + 14)?
The square root is approximately 27.258.
To find the square root, we need to find the sum of (729 + 14). 729 + 14 = 743, and then √743 ≈ 27.258.
Therefore, the square root of (729 + 14) is approximately ±27.258.
Find the perimeter of the rectangle if its length ‘l’ is √743 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle as approximately 94.516 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√743 + 20) ≈ 2 × (27.258 + 20) ≈ 2 × 47.258 ≈ 94.516 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.
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