126 LearnersLast updated on May 26th, 2025
Square Root of 266
If a number is multiplied by itself, the result is a square. The inverse of squaring is finding a square root. Square roots have applications in fields like vehicle design, finance, etc. Here, we will discuss the square root of 266.
What is the Square Root of 266?
The square root is the inverse of squaring a number. 266 is not a perfect square. The square root of 266 is expressed in both radical and exponential form. In radical form, it is written as √266, whereas in exponential form it is expressed as (266)^(1/2). √266 ≈ 16.30951, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 266
The prime factorization method is suitable for perfect square numbers. For non-perfect squares like 266, methods like the long-division method and approximation method are used. Let's explore these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 266 by Prime Factorization Method
The prime factorization of a number is the product of its prime factors. Let's see how 266 is broken down into its prime factors:
Step 1: Find the prime factors of 266. Breaking it down, we get 2 x 7 x 19.
Step 2: Since 266 is not a perfect square, its prime factors cannot be grouped into pairs.
Therefore, calculating √266 using prime factorization alone is not feasible.
Square Root of 266 by Long Division Method
The long division method is particularly used for non-perfect squares. In this method, we find the closest perfect square number to the given number. Let's learn how to find the square root using the long division method, step by step:
Step 1: Begin by grouping the digits from right to left. For 266, group as 66 and 2.
Step 2: Find n whose square is ≤ 2. Here, n is 1 because 1 x 1 = 1 ≤ 2. Subtract 1 from 2 to get a remainder of 1.
Step 3: Bring down 66 to make the new dividend 166.
Step 4: Double the divisor (1) to get 2.
Step 5: Find n such that 2n x n ≤ 166. n is 6 because 26 x 6 = 156.
Step 6: Subtract 156 from 166 to get a remainder of 10.
Step 7: Add a decimal point, bring down two zeros to make the new dividend 1000.
Step 8: Find n for 326n x n ≤ 1000. n is 3 because 326 x 3 = 978.
Step 9: Subtract 978 from 1000 to get 22.
Step 10: The quotient is 16.3.
Step 11: Continue until two decimal places are found or until the remainder is zero.
So, √266 ≈ 16.31.
Square Root of 266 by Approximation Method
The approximation method is another way to find square roots, providing an easy way to estimate the square root of a number. Let's use this method for 266:
Step 1: Identify the closest perfect squares around 266. The smallest perfect square is 256 (16^2) and the largest is 289 (17^2). √266 falls between 16 and 17.
Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Applying the formula: (266 - 256) / (289 - 256) = 10 / 33 = 0.303. Adding this to the lower bound, 16 + 0.303 ≈ 16.303.
Thus, the square root of 266 is approximately 16.31.
Common Mistakes and How to Avoid Them in the Square Root of 266
Students often make errors when finding square roots, such as forgetting about the negative root or skipping steps in long division. Let's look at some common mistakes in detail.
Mistake 1
Forgetting about the negative square root
Students should be aware that numbers have both positive and negative square roots. However, we usually consider only the positive square root for practical purposes.
For example, √50 = 7.07; there is also -7.07, which should not be overlooked.
Mistake 2
Not adding the square root symbol in the proper places
Incorrect use of the square root symbol is a common mistake. Simplifying numbers inside the square root before proceeding is crucial.
For example, √(12+24) ≠ √12 + √24. The correct expression is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
Students must simplify numbers inside the square root to identify the approximate value accurately.
For example, √50 = 7 is incorrect; the correct value is √50 ≈ 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
It's essential to distinguish between cube roots and square roots.
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. Students may skip steps accidentally, leading to incorrect results. This can occur during subtraction or other steps.
Square Root of 266 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √266?
The area of the square is approximately 706.57 square units.
Explanation
The area of a square = side^2.
The side length is given as √266.
Area = side^2 = (√266) x (√266) = 266.
Therefore, the area of the square box is approximately 706.57 square units.
Problem 2
A square-shaped building measuring 266 square feet is built; if each of the sides is √266, what will be the square feet of half of the building?
133 square feet
Explanation
Since the building is square-shaped, divide 266 by 2 to find half the area. 266 / 2 = 133.
So, half of the building measures 133 square feet.
Problem 3
Calculate √266 x 4.
Approximately 65.24
Explanation
First, find the square root of 266, which is approximately 16.31.
Then multiply 16.31 by 4. 16.31 x 4 ≈ 65.24.
Problem 4
What will be the square root of (256 + 10)?
The square root is approximately 16.12.
Explanation
To find the square root, find the sum of 256 + 10 = 266.
The square root of 266 is approximately 16.31.
Therefore, the square root of (256 + 10) is approximately ±16.31.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √266 units and the width ‘w’ is 40 units.
The perimeter of the rectangle is approximately 112.62 units.
Explanation
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√266 + 40)
≈ 2 × (16.31 + 40).
Perimeter ≈ 2 × 56.31 = 112.62 units.
FAQ on Square Root of 266
1.What is √266 in its simplest form?
2.Mention the factors of 266.
3.Calculate the square of 266.
4.Is 266 a prime number?
5.266 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 266?
8.How do technology and digital tools in United States support learning Algebra and Square Root of 266?
9.Does learning Algebra support future career opportunities for students in United States?
Important Glossaries for the Square Root of 266
- Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse operation is finding the square root: √16 = 4.
- Irrational number: An irrational number cannot be expressed as a simple fraction of two integers.
- Decimal: A decimal is a number that contains a whole number and a fractional part separated by a decimal point, like 16.31.
- Prime factorization: The process of expressing a number as the product of its prime factors.
- Perfect square: A number that is the square of an integer. For example, 256 is a perfect square because it is 16^2.
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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.
