123 LearnersLast updated on May 26th, 2025
Square Root of 465
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.
What is 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.
Finding the Square Root of 465
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:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 465 by Prime Factorization Method
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.
Square Root of 465 by Long Division Method
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.
Square Root of 465 by Approximation Method
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.
Common Mistakes and How to Avoid Them in the Square Root of 465
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.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example: √50 = 7.07, but there is also -7.07, which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important before moving on to other steps.
For example: √(12+24) ≠ √12 + √24. The correct simplification is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example: √50 ≈ 7.071, not 7.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube roots and square roots constantly as this will help them avoid mistakes.
For example: √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Due to this, many students might skip a step accidentally, which results in a wrong answer at the end. This may happen in subtraction or elsewhere.
Square root of 465 Examples
Problem 1
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.
Explanation
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.
Problem 2
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
Explanation
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.
Problem 3
Calculate √465 x 5.
107.819
Explanation
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.
Problem 4
What will be the square root of (435 + 30)?
The square root is approximately 22.
Explanation
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.
Problem 5
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.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√465 + 38) = 2 × (21.5638 + 38) ≈ 2 × 59.5638 ≈ 119.13 units.
FAQ on Square Root of 465
1.What is √465 in its simplest form?
2.Mention the factors of 465.
3.Calculate the square of 465.
4.Is 465 a prime number?
5.465 is divisible by?
6.How does learning Algebra help students in Qatar make better decisions in daily life?
7.How can cultural or local activities in Qatar support learning Algebra topics such as Square Root of 465?
8.How do technology and digital tools in Qatar support learning Algebra and Square Root of 465?
9.Does learning Algebra support future career opportunities for students in Qatar?
Important Glossaries for the Square Root of 465
- Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, 4^2 = 16, and the square root of 16 is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written as a simple fraction (p/q), where q is not equal to zero and p and q are integers.
- Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 16 is 4, although -4 is also a square root of 16.
- Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
- Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point. For example, 7.86, 8.65, and 9.42 are decimals.
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Jaskaran Singh Saluja
About the Author
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
