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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 461.
The square root is the inverse of the square of the number. 461 is not a perfect square. The square root of 461 is expressed in both radical and exponential form. In the radical form, it is expressed as √461, whereas (461)^(1/2) in the exponential form. √461 ≈ 21.472, 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 461 is broken down into its prime factors.
Step 1: Finding the prime factors of 461 Breaking it down, we see that 461 is already a prime number and cannot be broken down into smaller prime factors.
Step 2: Since 461 is a prime number, it cannot be grouped in pairs.
Therefore, calculating 461 using prime factorization does not yield a simple result.
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 digits of 461 from right to left. In the case of 461, we do not need to group as it is a three-digit number.
Step 2: Now we need to find n whose square is less than or equal to 4. We can say n is 2 because 2^2 is 4, which is exactly equal to 4. Now the quotient is 2, and the remainder is 0 after subtracting 4 from 4.
Step 3: Now let us bring down 61, which is the new dividend. 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; we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 61. Let us consider n as 1; now 41 × 1 = 41.
Step 6: Subtract 61 from 41; the difference is 20.
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 2000.
Step 8: Now we need to find the new divisor that is 42n. Consider n as 4; we get 424 × 4 = 1696.
Step 9: Subtracting 1696 from 2000 gives us the remainder 304.
Step 10: Now the quotient is 21.4.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.
So the square root of √461 is approximately 21.47.
The approximation method is another method for finding the 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 461 using the approximation method.
Step 1: Now we have to find the closest perfect square of √461.
The smallest perfect square less than 461 is 400, and the largest perfect square greater than 461 is 484. √461 falls somewhere between 20 and 22.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).
Going by the formula (461 - 400) ÷ (484 - 400) = 61/84 ≈ 0.726. 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 + 0.726 = 21.726, so the square root of 461 is approximately 21.726.
Students do make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. 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 √461?
The area of the square is approximately 461 square units.
The area of the square = side^2.
The side length is given as √461.
Area of the square = side^2 = √461 × √461 = 461.
Therefore, the area of the square box is approximately 461 square units.
A square-shaped building measuring 461 square feet is built; if each of the sides is √461, what will be the square feet of half of the building?
230.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 461 by 2 gives us 230.5.
So half of the building measures 230.5 square feet.
Calculate √461 × 5.
Approximately 107.36
The first step is to find the square root of 461, which is approximately 21.47.
The second step is to multiply 21.47 by 5.
So 21.47 × 5 ≈ 107.36.
What will be the square root of (441 + 20)?
The square root is approximately 21.47
To find the square root, we need to find the sum of (441 + 20). 441 + 20 = 461, and then √461 ≈ 21.47.
Therefore, the square root of (441 + 20) is approximately ±21.47.
Find the perimeter of the rectangle if its length ‘l’ is √461 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 118.94 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√461 + 38) ≈ 2 × (21.47 + 38) ≈ 2 × 59.47 ≈ 118.94 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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