144 LearnersLast updated on May 26th, 2025
Square Root of 105
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 like vehicle design, finance, etc. Here, we will discuss the square root of 105.
What is the Square Root of 105?
The square root is the inverse of the square of the number. 105 is not a perfect square. The square root of 105 is expressed in both radical and exponential form.
In the radical form, it is expressed as √105, whereas (105)(1/2) in the exponential form. √105 ≈ 10.24695, 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 105
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:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 105 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 105 is broken down into its prime factors.
Step 1: Finding the prime factors of 105. Breaking it down, we get 3 x 5 x 7: 31 x 51 x 71
Step 2: Now we found out the prime factors of 105. The second step is to make pairs of those prime factors. Since 105 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.
Therefore, calculating √105 using prime factorization is impossible.
Square Root of 105 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 105, we need to group it as 05 and 1.
Step 2: Now we need to find n whose square is 1. We can say n is ‘1’ because 1 x 1 is lesser than or equal to 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.
Step 3: Now let us bring down 05, which is the new dividend. Add the old divisor with the same number, 1 + 1, we get 2, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.
Step 5: The next step is finding 2n x n ≤ 05. Since 2 x 1 x 1 = 2, it's not possible. We continue with 2 x 0 x 0 = 0.
Step 6: Subtract 05 from 0, the difference is 5, and the quotient is 10.
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 500.
Step 8: Now we need to find the new divisor that is 20 because 202 x 2 = 404. Step 9: Subtracting 404 from 500, we get the result 96.
Step 10: Now the quotient is 10.2.
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 √105 is approximately 10.25.
Square Root of 105 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. Now let us learn how to find the square root of 105 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √105. The smallest perfect square less than 105 is 100, and the largest perfect square greater than 105 is 121. √105 falls somewhere between 10 and 11.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula, (105 - 100) ÷ (121 - 100) = 5/21 ≈ 0.238.
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 10 + 0.238 = 10.238, so the square root of 105 is approximately 10.238.
Common Mistakes and How to Avoid Them in the Square Root of 105
Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping long division methods, etc. Now let us 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, there is also -7.07 which should not be forgotten.
Square Root of 105 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √105?
The area of the square is 110.25 square units.
Explanation
The area of the square = side².
The side length is given as √105.
Area of the square = side² = (√105)² = 10.25 × 10.25 = 105.0625
Therefore, the area of the square box is approximately 110.25 square units.
Problem 2
A square-shaped building measuring 105 square feet is built; if each of the sides is √105, what will be the square feet of half of the building?
52.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 105 by 2 gives us 52.5.
So half of the building measures 52.5 square feet.
Problem 3
Calculate √105 x 5.
51.235
Explanation
The first step is to find the square root of 105, which is approximately 10.247.
The second step is to multiply 10.247 by 5.
So 10.247 x 5 = 51.235.
Problem 4
What will be the square root of (100 + 5)?
The square root is approximately 10.25.
Explanation
To find the square root, we need to find the sum of (100 + 5), which is 105.
The square root of 105 is approximately 10.25.
Therefore, the square root of (100 + 5) is ±10.25.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √105 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as 96.49 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√105 + 38)
= 2 × (10.25 + 38)
= 2 × 48.25
= 96.49 units.
FAQ on Square Root of 105
1.What is √105 in its simplest form?
2.Mention the factors of 105.
3.Calculate the square of 105.
4.Is 105 a prime number?
5.105 is divisible by?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of 105?
8.How do technology and digital tools in United States support learning Algebra and Square Root of 105?
9.Does learning Algebra support future career opportunities for students in United States?
Important Glossaries for the Square Root of 105
- Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
- Principal square root: A number has both positive and negative square roots; however, it is always a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
- Prime factorization: The process of breaking down a number into its basic building blocks of prime numbers. For example, the prime factorization of 105 is 3 x 5 x 7.
- Long division method: A method used to find the square root of non-perfect square numbers by dividing the number into pairs and calculating step by step.
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Jaskaran Singh Saluja
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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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