Step-by-Step Guide to Finding the Square Root of 225
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 for Students
1. What is the square root of 225?
The square root of 225 is 15. This means that 15 multiplied by itself equals 225, i.e., 15 × 15 = 225. In radical notation, it is written as √225 = 15. The result most often refers to the positive or principal square root.
2. How do you find the square root of 225 by division method?
To find the square root of 225 by the long division method, follow these steps:
1. Pair the digits from right to left (22)(5).
2. Find the largest number whose square is less than or equal to the first pair.
3. Subtract its square and bring down the next pair.
4. Double the quotient and find a digit to append such that when multiplied, it is less than or equal to the dividend.
5. Continue until the remainder is zero.
For 225, the result is 15.
3. Is 225 a perfect square?
Yes, 225 is a perfect square because it is the product of the integer 15 multiplied by itself (15 × 15 = 225). Perfect squares are numbers whose square roots are whole numbers.
4. Can I write the square root of 225 as a fraction?
Yes, the square root of 225 can be expressed as a fraction. Since √225 = 15, and 15 is a whole number, it can be written as 15/1, which is a rational fraction form.
5. What is the square root of 225 in decimal and radical form?
The square root of 225 in radical form is √225, which simplifies to 15. In decimal form, it is simply 15.0. Since 225 is a perfect square, the decimal is exact and whole.
6. What is the simplified form of the square root of 225?
The simplified form of the square root of 225 is 15, because 225 = 152. There are no radicals left after simplification since 225 is a perfect square.
7. Why do students confuse the square root of 225 with 2250 or 22500?
Students often confuse √225 with √2250 or √22500 due to the similarity in numbers. However, 225 is a perfect square with an exact root 15, while 2250 and 22500 are not perfect squares and their roots are irrational. Understanding perfect squares vs. non-perfect squares helps avoid this confusion.
8. Why is the step-by-step division method preferred in some exams?
The step-by-step division method is preferred because it breaks down the square root calculation into systematic steps that can be easily followed and verified during exams. It reduces errors and helps students show work methodically.
9. Can the square root of 225 ever be negative in exam answers?
Mathematically, the square root of 225 has two values: +15 and -15 because both squared equal 225. However, in most exam contexts, unless asked otherwise, the principal (positive) square root of 15 is the accepted answer since length and magnitude are positive quantities.
10. Explain common mistakes while simplifying square roots like 225/16.
A common mistake is not simplifying the square roots of numerator and denominator separately when dealing with expressions like √(225/16). The correct simplification is √225 ÷ √16 = 15 ÷ 4 = 15/4. Avoid taking the square root of the fraction as one whole without breaking it down.
11. Why is understanding perfect squares important for fast problem solving?
Understanding perfect squares helps students quickly recognize which numbers have whole number square roots. This speeds up mental calculations, reduces reliance on calculators, and improves accuracy during exams.
12. What are the different methods to find the square root of 225?
There are mainly three methods to find the square root of 225:
• Prime Factorization: Break 225 into prime factors and pair them.
• Long Division Method: Divide in pairs from right to left to find root stepwise.
• Repeated Subtraction: Subtract odd numbers consecutively until zero; the count is the square root.
These methods help understand the concept deeply and are useful depending on the exam pattern.
