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 465.
The square root is the inverse of the square of the number. 465 is not a perfect square. The square root of 465 is expressed in both radical and exponential forms. In radical form, it is expressed as √465, whereas in exponential form, it is expressed as (465)^(1/2). √465 ≈ 21.5638, 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, for non-perfect square numbers, the prime factorization method is not applicable, and instead, 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 465 is broken down into its prime factors.
Step 1: Finding the prime factors of 465
Breaking it down, we get 3 x 5 x 31: 3^1 x 5^1 x 31^1
Step 2: Now we found out the prime factors of 465. The second step is to make pairs of those prime factors. Since 465 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 465 using prime factorization is not feasible.
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 465, we need to group it as 65 and 4.
Step 2: Now we need to find n whose square is less than or equal to 4. We can say n as '2' because 2 x 2 = 4, which is less than or equal to 4. Now the quotient is 2, and after subtracting 4 - 4, the remainder is 0.
Step 3: Now let us bring down 65, which is the new dividend. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor.
Step 4: The new divisor will be 2n, so we need to find the value of n such that 2n x n ≤ 65. Let us consider n as 1, now 41 x 1 = 41.
Step 5: Subtract 41 from 65; the difference is 24, and the quotient is 21.
Step 6: 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 2400.
Step 7: Now we need to find the new divisor that is 423. Let's consider n as 5 because 425 x 5 = 2125.
Step 8: Subtracting 2125 from 2400, we get the result 275.
Step 9: Now the quotient is 21.5
Step 10: 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 √465 is approximately 21.56.
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 465 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √465. The smallest perfect square less than 465 is 441 (21^2), and the largest perfect square greater than 465 is 484 (22^2). √465 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). Using the formula: (465 - 441) / (484 - 441) = 24 / 43 ≈ 0.558.
Using the formula, we identified the decimal point of our square root.
The next step is adding the value we got initially to the integer number, which is 21 + 0.558 ≈ 21.558, so the square root of 465 is approximately 21.558.
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. Now let's 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 √465?
The area of the square is approximately 2165.89 square units.
The area of the square = side^2.
The side length is given as √465.
Area of the square = side^2 = √465 x √465 = 21.5638 x 21.5638 ≈ 2165.89.
Therefore, the area of the square box is approximately 2165.89 square units.
A square-shaped building measuring 465 square feet is built; if each of the sides is √465, what will be the square feet of half of the building?
232.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 465 by 2, we get 232.5.
So half of the building measures 232.5 square feet.
Calculate √465 x 5.
107.819
The first step is to find the square root of 465, which is approximately 21.5638.
The second step is to multiply 21.5638 by 5.
So 21.5638 x 5 ≈ 107.819.
What will be the square root of (435 + 30)?
The square root is approximately 22.
To find the square root, we need to find the sum of (435 + 30). 435 + 30 = 465, and then √465 ≈ 21.5638.
Therefore, the square root of (435 + 30) is approximately ±21.5638.
Find the perimeter of the rectangle if its length ‘l’ is √465 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 119.13 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√465 + 38) = 2 × (21.5638 + 38) ≈ 2 × 59.5638 ≈ 119.13 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.