Last updated on May 26th, 2025
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.
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.
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:
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.
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.
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.
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.
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.
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.
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
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.
Calculate √266 x 4.
Approximately 65.24
First, find the square root of 266, which is approximately 16.31.
Then multiply 16.31 by 4. 16.31 x 4 ≈ 65.24.
What will be the square root of (256 + 10)?
The square root is approximately 16.12.
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.
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.
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√266 + 40)
≈ 2 × (16.31 + 40).
Perimeter ≈ 2 × 56.31 = 112.62 units.
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.
: He loves to play the quiz with kids through algebra to make kids love it.