125 LearnersLast updated on May 26th, 2025
Square Root of 405
If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 405.
What is the Square Root of 405?
The square root is the inverse of the square of a number. 405 is not a perfect square. The square root of 405 is expressed in both radical and exponential form. In the radical form, it is expressed as √405, whereas (405)^(1/2) in the exponential form. √405 ≈ 20.1246, 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 405
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 405, 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 405 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 405 is broken down into its prime factors.
Step 1: Finding the prime factors of 405 Breaking it down, we get 3 x 3 x 3 x 3 x 5: 3^4 x 5
Step 2: Now we found the prime factors of 405. The second step is to make pairs of those prime factors. Since 405 is not a perfect square, the digits of the number can’t be grouped in pairs completely.
Therefore, calculating √405 using prime factorization alone does not yield an exact result.
Square Root of 405 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, group the numbers from right to left. In the case of 405, group it as 05 and 4.
Step 2: 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 the remainder is 0.
Step 3: Bring down 05, 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 4n. We need to find the value of n such that 4n x n ≤ 05. Let us consider n as 1, now 4 x 1 x 1 = 4.
Step 5: Subtract 5 from 4, the difference is 1.
Step 6: Since the dividend is less than the divisor, we add a decimal point and bring down two zeros, making the new dividend 100.
Step 7: Find the new divisor, which is 41. Use 41n x n ≤ 100. Let n be 2, then 41 x 2 x 2 = 164, which doesn't fit. Try n = 1, and 41 x 1 x 1 = 41.
Step 8: Subtract 41 from 100, the remainder is 59, and the quotient becomes 20.1.
Step 9: Continue these steps for more precision until you get the desired number of decimal places.
So the square root of √405 ≈ 20.12
Square Root of 405 by Approximation Method
The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 405 using the approximation method.
Step 1: Identify the closest perfect squares to √405. The nearest perfect squares are 400 and 441. √405 falls between 20 and 21.
Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (405 - 400) / (441 - 400) = 5 / 41 ≈ 0.122 Add the initial integer value to the decimal: 20 + 0.122 ≈ 20.122
Thus, the square root of 405 is approximately 20.12.
Common Mistakes and How to Avoid Them in the Square Root of 405
Students often 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 common mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important to remember that a number has both positive and negative square roots. However, we typically use the positive square root.
For example, √50 = ±7.071, but we usually focus on the positive value.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root symbol correctly is a common mistake. It's important to simplify numbers inside the square root before proceeding.
For example: √(12 + 24) ≠ √12 + √24. The correct method is √(12 + 24) = √36 = 6.
Mistake 3
Not finding the correct value of a non-perfect square
Students should simplify numbers inside the square root to avoid approximation errors.
For example, √50 ≠ 7; the correct value is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to distinguish between cube roots and square roots.
For example, √50 and ∛50 are different operations.
Mistake 5
Making mistakes in long division
The long division method involves many steps, so it's easy to skip a step, leading to incorrect answers. This mistake can occur in subtraction or other steps.
Square Root of 405 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √405?
The area of the square is 405 square units.
Explanation
The area of a square = side^2. The side length is given as √405.
Area = side^2 = √405 x √405 = 405
Therefore, the area of the square box is 405 square units.
Problem 2
A square-shaped building measuring 405 square feet is built; if each of the sides is √405, what will be the square feet of half of the building?
202.5 square feet
Explanation
We can divide the given area by 2 as the building is square-shaped.
Dividing 405 by 2 = 202.5 So half of the building measures 202.5 square feet.
Problem 3
Calculate √405 x 5.
100.62
Explanation
First, find the square root of 405, which is approximately 20.12.
Then multiply 20.12 by 5. 20.12 x 5 = 100.62
Problem 4
What will be the square root of (400 + 5)?
The square root is approximately 20.12
Explanation
To find the square root, calculate (400 + 5) = 405.
The square root of 405 is approximately 20.12.
Therefore, the square root of (400 + 5) is approximately ±20.12.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √405 units and the width ‘w’ is 30 units.
The perimeter of the rectangle is approximately 100.24 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√405 + 30) ≈ 2 × (20.12 + 30)
Perimeter ≈ 2 × 50.12 = 100.24 units.
FAQ on Square Root of 405
1.What is √405 in its simplest form?
2.Mention the factors of 405.
3.Calculate the square of 405.
4.Is 405 a prime number?
5.405 is divisible by?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of 405?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of 405?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of 405
- Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse is √16 = 4.
- Irrational number: An irrational number cannot be written in the form of p/q, where q is not zero, and p and q are integers.
- Principal square root: The non-negative square root of a number is called the principal square root, used primarily in real-world applications.
- Prime factor: A prime factor is a factor of a number that is a prime number itself. For example, the prime factors of 405 are 3 and 5.
- Approximation: The process of finding a value close to the actual value. For example, the approximate value of √405 is 20.12.
Explore More algebra
About BrightChamps in Australia
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.
