138 LearnersLast updated on May 26th, 2025
Square Root of 113
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 various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 113.
What is the Square Root of 113?
The square root is the inverse of the square of the number. 113 is not a perfect square. The square root of 113 is expressed in both radical and exponential form.
In the radical form, it is expressed as √113, whereas (113)(1/2) in the exponential form. √113 ≈ 10.63014581273465, 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 113
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 113 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 113 is broken down into its prime factors:
Step 1: Finding the prime factors of 113
113 is a prime number, so it cannot be broken down further.
This means the prime factorization of 113 is simply 113 itself.
Since 113 is not a perfect square, calculating √113 using prime factorization directly is not feasible.
Square Root of 113 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 113, we need to group it as 13 and 1.
Step 2: Now we need to find n whose square is less than or equal to 1. We can say n as '1' because 1 × 1 is less than or equal to 1. Now the quotient is 1, and the remainder is 0 after subtracting 1 from 1.
Step 3: Bring down 13, which is the new dividend. Add the old divisor with the same number, 1 + 1, to 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, and we need to find the value of n.
Step 5: The next step is finding 2n × n ≤ 13. Let us consider n as 6, now 26 × 6 = 156.
Step 6: Since 156 is greater than 13, we try n as 5, so 25 × 5 = 125, which is still greater. We try n as 4, so 24 × 4 = 96, which is less than 113.
Step 7: Subtract 96 from 113 to get the difference, which is 17. The quotient is 10.6.
Step 8: 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 1700.
Step 9: Continue this process until you reach the desired precision. The square root of √113 is approximately 10.63.
Square Root of 113 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 113 using the approximation method.
Step 1: Now we have to find the closest perfect square to √113. The smallest perfect square less than 113 is 100, and the largest perfect square greater than 113 is 121. √113 falls somewhere between 10 and 11.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (113 - 100) ÷ (121 - 100) ≈ 0.619. 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 + 0.63 ≈ 10.63, so the square root of 113 is approximately 10.63.
Common Mistakes and How to Avoid Them in the Square Root of 113
Students do make mistakes while finding the square root, such as 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.
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: √113 = 10.63, there is also -10.63 which should not be forgotten.
Square Root of 113 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √113?
The area of the square is 113 square units.
Explanation
The area of the square = side².
The side length is given as √113.
Area of the square = side² = √113 × √113 = 113.
Therefore, the area of the square box is 113 square units.
Problem 2
A square-shaped building measuring 113 square feet is built; if each of the sides is √113, what will be the square feet of half of the building?
56.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 113 by 2 = we get 56.5.
So half of the building measures 56.5 square feet.
Problem 3
Calculate √113 × 5.
53.15
Explanation
The first step is to find the square root of 113 which is approximately 10.63. The second step is to multiply 10.63 by 5. So 10.63 × 5 ≈ 53.15.
Problem 4
What will be the square root of (100 + 13)?
The square root is 10.63
Explanation
To find the square root, we need to find the sum of (100 + 13). 100 + 13 = 113, and then √113 ≈ 10.63.
Therefore, the square root of (100 + 13) is approximately 10.63.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √113 units and the width ‘w’ is 10 units.
We find the perimeter of the rectangle as 41.26 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√113 + 10)
= 2 × (10.63 + 10)
= 2 × 20.63 = 41.26 units.
FAQ on Square Root of 113
1.What is √113 in its simplest form?
2.Mention the factors of 113.
3.Calculate the square of 113.
4.Is 113 a prime number?
5.113 is divisible by?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 113?
8.How do technology and digital tools in India support learning Algebra and Square Root of 113?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 113
- Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and the square root of 16 is √16 = 4.
- Irrational number: An irrational number cannot be expressed as a simple fraction or ratio of two integers. It has a non-repeating, non-terminating decimal expansion.
- Principal square root: The principal square root is the non-negative square root of a number. Even though a number has both positive and negative square roots, the principal square root is the positive one.
- Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 113 is an example of a prime number.
- Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point. For example, 10.63 is a decimal.
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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.
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