How to Find the Square Root of 225 Step by Step with Formula and Examples
The concept of square root of 225 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding how to find and simplify the square root of 225 is a valuable skill for students preparing for board exams and various entrance tests.
Understanding Square Root of 225
A square root refers to any number which, when multiplied by itself, gives the original number. Hence, the square root of 225 is a value that, when squared, becomes 225. This concept is widely used in algebra, geometry, and number theory. It is also crucial for solving area-based and quadratic problems in mathematics. The square root is often written in radical notation as √225, in exponential form as 2251/2, and can be expressed in both fraction and decimal forms.
Is 225 a Perfect Square?
Yes, 225 is a perfect square.
Here’s a simple chart of common perfect squares for better clarity:
| Number | Square Root | Perfect Square? |
|---|---|---|
| 144 | 12 | Yes |
| 169 | 13 | Yes |
| 225 | 15 | Yes |
| 256 | 16 | Yes |
How to Find the Square Root of 225
There are several methods to calculate the square root of 225. Here are the two most commonly used approaches:
1. Prime Factorization Method
Pair the same numbers: (3 × 3) and (5 × 5)
Take the square root of each pair:
√225 = √(3 × 3 × 5 × 5) = √(3 × 3) × √(5 × 5) = 3 × 5 = 15
2. Long Division Method
Step 2. Find a number whose square is ≤ 2. That is 1, because 1 × 1 = 1.
Step 3. Subtract 1 from 2. Remainder: 1. Bring down next pair (25), making 125.
Step 4. Double quotient so far (which is 1). Write 2_.
Step 5. Find a digit x such that (20 + x) × x ≤ 125. Try x = 5: (20+5)×5=125.
Step 6. Subtract 125 from 125. Remainder: 0.
The answer is 15.
Square Root of 225 in Radical, Fraction, and Decimal Forms
Here is a table showing the square root of 225 in different mathematical forms for complete clarity:
| Form | Representation | Value |
|---|---|---|
| Radical | √225 | 15 |
| Fraction | 15/1 | 15 |
| Decimal | 15.0 | 15 |
Worked Example – Solving a Problem
Let’s see a step-by-step example for a real-world question based on square root of 225:
Step 1. Area of square = side × side = a²
Step 2. a² = 225
Step 3. a = √225
Step 4. a = 15
So, each side is 15 m.
Comparison with Related Square Roots
It's helpful to compare the square root of 225 with nearby perfect squares and related numbers:
| Number | Square Root |
|---|---|
| 144 | 12 |
| 169 | 13 |
| 225 | 15 |
| 256 | 16 |
| 289 | 17 |
| 225/16 | 15/4 = 3.75 |
| 2250 | ≈ 47.434 |
Common Mistakes to Avoid
- Assuming only positive values for the square root (It can be ±15, but in most school problems, only the positive or principal square root is required.)
- Mixing up 225 with 2250 or 22500 when reading or simplifying roots.
- Forgetting to pair prime factors when using prime factorization.
Real-World Applications
The square root of 225 is used in construction, engineering, and area calculations. For instance, finding the dimensions of a square plot given its area. In competitive exams, square roots like √225 help in quick estimation and calculating directly. Vedantu helps students master these concepts to solve practical and theoretical problems confidently.
We explored the idea of square root of 225, different calculation methods, and how it applies to real-world and mathematical problems. Practicing these steps with Vedantu resources builds a thorough understanding of square roots and their relevance in exams and everyday life.
To expand your understanding of related concepts, explore these helpful resources:
- Square root of 144
- Square root of 256
- Square root of 289
- Factors of 225
- Square root finder
- Prime numbers
- Estimating square root and cube root
- Square root and cube root
- How to find square root of a number
- Square root table
FAQs on Square Root of 225 Explained with Solution
1. What is the square root of 225?
The square root of 225 is 15. This is because 15 × 15 = 225, so 15 is the number which, when multiplied by itself, gives 225. Although both 15 and −15 satisfy the equation x² = 225, the term principal square root refers to the positive value, which is 15.
2. How do you find the square root of 225 step by step?
You can find the square root of 225 by prime factorization or recognizing it as a perfect square; the result is 15.
- Step 1: Prime factorize 225 = 3 × 3 × 5 × 5.
- Step 2: Pair the equal factors: (3 × 3) and (5 × 5).
- Step 3: Take one factor from each pair: 3 × 5 = 15.
3. Is 225 a perfect square?
Yes, 225 is a perfect square because it can be written as 15². A perfect square is a number obtained by squaring a whole number, and since 15 × 15 = 225, 225 satisfies this definition.
4. What is the principal square root of 225?
The principal square root of 225 is 15. The principal square root always refers to the positive value of a number whose square equals the given number. Although −15 also squares to 225, the principal square root is positive.
5. What are the positive and negative square roots of 225?
The positive and negative square roots of 225 are +15 and −15. Both values satisfy the equation x² = 225 because:
- 15 × 15 = 225
- (−15) × (−15) = 225
6. What is √225 in simplest radical form?
√225 in simplest radical form is 15. Since 225 is a perfect square (15²), the radical simplifies completely without leaving any number inside the square root symbol.
7. How do you check if 15 is the square root of 225?
You can check by squaring 15; since 15² = 225, 15 is the square root of 225.
- Multiply 15 × 15.
- The result is 225.
8. What is the square root of 225 using the long division method?
Using the long division method, the square root of 225 is 15.
- Group digits in pairs from right: 2 | 25.
- Find the largest number whose square is ≤ 2, which is 1.
- Proceed with division and subtraction steps to get the next digit 5.
9. What is the value of 225 raised to the power 1/2?
225 raised to the power 1/2 equals 15. The exponent 1/2 represents the square root, so 225^(1/2) = √225 = 15.
10. What are some real-life examples of the square root of 225?
The square root of 225, which is 15, appears in real-life measurements involving area and geometry.
- If a square has an area of 225 square units, each side measures 15 units.
- If a garden covers 225 m² in a square shape, each side is 15 m long.
