125 LearnersLast updated on May 26th, 2025
Square Root of 226
If a number is multiplied by itself, the result is a square. The inverse operation, finding a square root, is used in various fields such as vehicle design and finance. Here, we will discuss the square root of 226.
What is the Square Root of 226?
The square root is the inverse operation of squaring a number. Since 226 is not a perfect square, its square root is expressed in both radical and exponential forms. In radical form, it is expressed as √226, whereas in exponential form it is expressed as (226)^(1/2). √226 ≈ 15.0333, 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 226
For perfect squares, the prime factorization method is typically used. However, for non-perfect squares like 226, methods such as long division and approximation are employed. Let us delve into these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 226 by Prime Factorization Method
The prime factorization method involves expressing a number as a product of its prime factors. Let's break down 226 into its prime factors:
Step 1: Finding the prime factors of 226 Breaking it down, we get 2 x 113: 2^1 x 113^1
Step 2: Since 226 is not a perfect square, the digits cannot be paired.
Therefore, calculating √226 using prime factorization is not feasible.
Square Root of 226 by Long Division Method
The long division method is well-suited for non-perfect squares. This method involves finding the closest perfect square for the given number. Let's find the square root using this method, step by step:
Step 1: Group the digits in pairs from right to left. For 226, consider it as 26 and 2.
Step 2: Find the largest number whose square is less than or equal to 2. Here, n = 1. Subtract 1^2 from 2, leaving a remainder of 1.
Step 3: Bring down the next pair, 26, making it 126. Double the divisor (1), giving 2, to form the new divisor.
Step 4: Find n such that 2n x n ≤ 126. Using n = 5, we have 25 x 5 = 125.
Step 5: Subtract 125 from 126, leaving a remainder of 1. The quotient is 15.
Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros, making it 100.
Step 7: The new divisor becomes 300 (from 2 x 15), and n is 3 since 303 x 3 = 909.
Step 8: Subtract 909 from 1000, leaving 91.
Step 9: Continue this process to achieve two decimal places.
Thus, √226 ≈ 15.03.
Square Root of 226 by Approximation Method
The approximation method provides an easy way to find the square root of a number. Here's how to find the square root of 226 using this method:
Step 1: Identify the closest perfect squares to √226. The nearest perfect square less than 226 is 225, and the nearest greater is 256. So, √226 is between 15 and 16.
Step 2: Apply the formula: (Given number - smaller perfect square)/(Greater perfect square - smaller perfect square). Using the formula (226 - 225) ÷ (256 - 225) = 1/31 ≈ 0.0323. Add this to the smaller perfect square root: 15 + 0.0323 ≈ 15.0323, so √226 ≈ 15.03.
Common Mistakes and How to Avoid Them in the Square Root of 226
Mistakes often occur when finding square roots, such as neglecting the negative square root or misapplying methods like long division. Let's explore common errors and how to avoid them.
Mistake 1
Forgetting about the negative square root
Students should remember that numbers have both positive and negative square roots. However, we typically use the positive square root.
For example, √50 = 7.07, but there is also -7.07.
Mistake 2
Not adding the square root symbol in proper places
Incorrectly applying the square root symbol is a common error. Students should simplify numbers inside the square root first.
For example: √(12 + 24) ≠ √12 + √24. The correct approach is √(12 + 24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
It's crucial to simplify numbers under the square root to find the approximate value.
For example, √50 ≠ 7; the accurate value is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students must understand the difference between cube roots and square roots.
For example: √50 and ∛50 are distinct.
Mistake 5
Making mistakes in long division
The long division method requires careful attention to detail. Students may skip steps, leading to incorrect results, often because of errors in subtraction or elsewhere.
Square Root of 226 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √226?
The area of the square is approximately 510.9929 square units.
Explanation
The area of a square = side².
With a side length of √226,
area = (√226)²
= 226 square units.
Problem 2
A square-shaped building measuring 226 square feet is built. If each side is √226, what will be the square feet of half the building?
113 square feet
Explanation
For a square building, dividing the area by 2 gives half the building's area.
226 ÷ 2 = 113 square feet.
Problem 3
Calculate √226 x 5.
Approximately 75.1665
Explanation
First, find √226 ≈ 15.0333, then multiply by 5: 15.0333 x 5 ≈ 75.1665.
Problem 4
What is the square root of 225 + 1?
The square root is 15.0333
Explanation
Calculate 225 + 1 = 226,
so √226 ≈ 15.0333.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √226 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 106.0666 units.
Explanation
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√226 + 38)
≈ 2 × (15.0333 + 38)
≈ 106.0666 units.
FAQ on Square Root of 226
1.What is √226 in its simplest form?
2.Mention the factors of 226.
3.Calculate the square of 226.
4.Is 226 a prime number?
5.226 is divisible by?
6.How does learning Algebra help students in Qatar make better decisions in daily life?
7.How can cultural or local activities in Qatar support learning Algebra topics such as Square Root of 226?
8.How do technology and digital tools in Qatar support learning Algebra and Square Root of 226?
9.Does learning Algebra support future career opportunities for students in Qatar?
Important Glossaries for the Square Root of 226
- Square root: The 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. For instance, √226 is irrational.
- Principal square root: The positive square root of a number is known as the principal square root.
- Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 226 is 2 x 113.
- Long division method: A technique used to find the square root of non-perfect squares by dividing them systematically.
Explore More algebra
About BrightChamps in Qatar
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.
