127 LearnersLast updated on May 26th, 2025
Square Root of 207
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 such as vehicle design, finance, etc. Here, we will discuss the square root of 207.
What is the Square Root of 207?
The square root is the inverse of the square of the number. 207 is not a perfect square. The square root of 207 is expressed in both radical and exponential form. In the radical form, it is expressed as √207, whereas in the exponential form it is expressed as (207)^(1/2). √207 ≈ 14.387, 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 207
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division and approximation methods are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 207 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 207 is broken down into its prime factors:
Step 1: Finding the prime factors of 207 Breaking it down, we get 3 x 3 x 23: 3^2 x 23
Step 2: Now we found out the prime factors of 207. The second step is to make pairs of those prime factors. Since 207 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.
Therefore, calculating √207 using prime factorization is not straightforward.
Square Root of 207 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 207, we need to group it as 07 and 2.
Step 2: Now we need to find n whose square is less than or equal to 2. We can say n is ‘1’ because 1 x 1 = 1, which is less than or equal to 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.
Step 3: Bring down 07, so the new dividend is 107. 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 2n. We need to find the value of n such that 2n x n is less than or equal to 107. Let us consider n as 4, now 24 x 4 = 96.
Step 5: Subtract 96 from 107, and the difference is 11. The quotient is 14.
Step 6: Since the dividend is still 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 1100.
Step 7: Now we need to find the new divisor, which is 287 because 287 x 4 = 1148.
Step 8: Subtracting 1148 from 1100 gives -48, but since it is negative, we adjust to get 1100 minus 108 = 992.
Step 9: Now the quotient is 14.3.
Step 10: Continue doing these steps until we get two numbers after the decimal point or the remainder is zero.
So the approximate square root of √207 is 14.39.
Square Root of 207 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 207 using the approximation method.
Step 1: Now we have to find the closest perfect squares to √207. The closest perfect squares to 207 are 196 (14^2) and 225 (15^2). √207 falls somewhere between 14 and 15.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (207 - 196) / (225 - 196) = 11/29 ≈ 0.379 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: 14 + 0.379 ≈ 14.379, so the square root of 207 is approximately 14.379.
Common Mistakes and How to Avoid Them in the Square Root of 207
Students do make mistakes while finding the square root, like 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, for many practical applications, we only consider the positive square root.
For example: √50 = 7.07, 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) = √36 is correct, whereas √12 + √24 is incorrect.
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 is incorrect; the correct answer is √50 ≈ 7.071067.
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 of the sum. This may happen during subtraction or elsewhere.
Square Root of 207 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √207?
The area of the square is approximately 207 square units.
Explanation
The area of the square = side^2.
The side length is given as √207.
Area of the square = side^2 = √207 x √207 ≈ 207.
Therefore, the area of the square box is approximately 207 square units.
Problem 2
A square-shaped building measuring 207 square feet is built; if each of the sides is √207, what will be the square feet of half of the building?
Approximately 103.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 207 by 2 gives approximately 103.5.
So half of the building measures approximately 103.5 square feet.
Problem 3
Calculate √207 x 5.
Approximately 71.935
Explanation
The first step is to find the square root of 207, which is approximately 14.39.
The second step is to multiply 14.39 by 5. So 14.39 x 5 ≈ 71.935.
Problem 4
What will be the square root of (207 + 18)?
The square root is 15.
Explanation
To find the square root, we need to find the sum of (207 + 18). 207 + 18 = 225, and then √225 = 15.
Therefore, the square root of (207 + 18) is ±15.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √207 units and the width ‘w’ is 20 units.
The perimeter of the rectangle is approximately 88.78 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√207 + 20) ≈ 2 × (14.39 + 20) ≈ 2 × 34.39 ≈ 88.78 units.
FAQ on Square Root of 207
1.What is √207 in its simplest form?
2.Mention the factors of 207.
3.Calculate the square of 207.
4.Is 207 a prime number?
5.207 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 207?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of 207?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of 207
- Square root: A square root is the inverse of a square. For example: 4^2 = 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, the positive square root is more commonly used in real-world applications. This is known as the principal square root.
- Prime factorization: The process of expressing a number as a product of its prime factors.
- Long division method: A method used to find the square root of a number, especially non-perfect squares, by breaking it down into smaller and more manageable divisions.
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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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