Why is 169 a Perfect Square?
The concept of square root of 169 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding how to find and use square roots makes algebra, geometry, and number properties much easier—especially for board exams.
Understanding Square Root of 169
A square root of 169 refers to a number that, when multiplied by itself, equals 169. It is commonly written as √169. This concept is widely used in perfect square numbers, simplifying radicals, and factorization. Since 169 is a perfect square, finding its square root is direct and often appears in maths exams and practical problems.
Quick Value and Definition
Square root of 169 is 13. Because 13 × 13 = 169, the value of √169 = 13. While mathematically there are both positive and negative square roots, in most school-level mathematics, we take the principal (positive) root unless otherwise stated.
Square Root of 169 in Different Forms
The square root of 169 can be written in several ways:
2. Exponential form: 169½
3. Simplest form: 13
169 is called a perfect square since its square root is a whole number. Recognising these helps students quickly solve square root problems in exams.
Step-by-Step Methods to Find the Square Root of 169
You can find the square root of 169 using different methods. Here are the most common:
- Express 169 as a product of its prime factors.
169 = 13 × 13
- Pair the same factors: (13, 13)
- Take one number from each pair to get the square root:
√169 = 13
2. Long Division Method
- Group digits in pairs from right to left. For 169: (1)(69)
- 1 × 1 = 1 (subtract 1 from 1 = 0; bring down 69)
- Double the divisor (1) to get 2_.
- Find a digit (3) such that 23 × 3 = 69.
- Subtract: 69 - 69 = 0. So, quotient is 13.
- Thus, √169 = 13
3. Repeated Subtraction Method (for perfect squares)
- Successively subtract odd numbers from 169 until the remainder is zero.
169-1=168
168-3=165
165-5=160
160-7=153
153-9=144
144-11=133
133-13=120
120-15=105
105-17=88
88-19=69
69-21=48
48-23=25
25-25=0
- Count the steps: 13 times
- Therefore, √169 = 13
Worked Example – Solving a Problem
Let’s find the square root of 169 step-by-step using the prime factorization method:
2. Since there are two 13s, make a pair: (13, 13)
3. The square root will be one number from the pair: 13
Final Answer: √169 = 13
Square Roots of Similar Perfect Squares
Here’s a helpful table to compare square root of 169 with other perfect squares near it:
Square Roots of Nearby Numbers
| Number | Square Root | Is it a Perfect Square? |
|---|---|---|
| 144 | 12 | Yes |
| 169 | 13 | Yes |
| 196 | 14 | Yes |
| 225 | 15 | Yes |
| 289 | 17 | Yes |
This table shows how the pattern of perfect squares helps remember common roots for quick calculations.
Common Mistakes to Avoid
- Forgetting that while √169 = ±13 mathematically, standard school answers use the positive root unless otherwise specified.
- Confusing 169 with 196 (which is 14 × 14, not 13 × 13).
- Trying to use the division method when not grouping digits correctly.
Practice Problems
- Find the square root of 16900.
- Simplify: √169 × √4.
- Write the square root of 169 as a decimal (to the nearest tenth).
- Which is greater: √169 or √225?
- Find the square root of 169 using the long division method (show all steps).
Real-World Applications
The concept of square root of 169 appears in daily life—such as calculating areas, side lengths in geometry, or solving quadratic equations. It also comes up in science and engineering. Vedantu helps students practice these concepts using board exam-style questions and everyday math problems.
Quick Revision Tips
- 13 × 13 = 169, so √169 = 13.
- 169 is a perfect square—memorise square roots of 1–20 for exams.
- Use prime factorization for any number—pair the factors for extracting square roots.
- For larger numbers (like 16900), first break the number into 169 × 100: √16900 = √169 × √100 = 13 × 10 = 130.
- Tables of perfect squares are helpful for last-minute revision.
Further Practice and Related Topics
- Square Root of 144
- Square Root of 196
- Square Root of 289
- Square Root Finder
- Factors of 169
- Prime Factorization
- Square Numbers
- Square Root Table
- Square Root Questions
- Squares and Square Roots
We explored the idea of square root of 169, methods to solve for it, common mistakes, and related board exam tricks. Practising these strategies with Vedantu will help you answer any square root question in exams confidently and efficiently.
FAQs on Square Root of 169: Explained with Simple Steps
1. What is the square root of 169?
The square root of 169 is 13 because 13 multiplied by itself equals 169, i.e., 13 × 13 = 169. This means that 13 is the principal square root, often denoted as √169 = 13.
2. How do you find the square root of 169?
You can find the square root of 169 using multiple methods such as:
1. Prime factorization: Break 169 into its prime factors (13 × 13) and take one factor.
2. Division method: Divide 169 into pairs and find a divisor step-by-step.
3. Repeated subtraction: Subtract successive odd numbers until reaching zero, the count equals the square root.
These methods help verify that √169 = 13.
3. Why is 169 a perfect square?
169 is a perfect square because it can be expressed as the square of a whole number. Specifically, 169 = 13 × 13, where 13 is an integer. Perfect squares always have integer square roots, making 169 part of this category.
4. How to write the square root of 169 in radical form?
The square root of 169 in radical form is written as √169. It represents the value that when squared equals 169, which simplifies to 13 since √169 = √(13 × 13) = 13.
5. What is the square root of 169 by prime factorization?
Using prime factorization, 169 is factored into its prime elements: 169 = 13 × 13. Taking one 13 gives the square root. Therefore, √169 = 13. This method is helpful for understanding the root via divisors.
6. Why is the square root of 169 not negative in board exams?
In board exams, the square root of 169 is generally taken as the principal (non-negative) root, which is 13. Although mathematically both +13 and -13 satisfy the equation, exam standards prefer the positive root to avoid confusion, unless specifically asked for both.
7. Why do students confuse 169 with 196 in square roots?
Students often confuse 169 with 196 because both are perfect squares and have similar digits. However, √169 = 13, and √196 = 14. Careful recognition of digits and memorizing common square numbers can help reduce this confusion.
8. How does writing in simplest radical form help in exams?
Writing the square root in simplest radical form helps because it:
• Shows clear understanding of simplifying radicals.
• Helps in solving complex problems faster.
• Is often required by board exam marking schemes.
For example, √169 simplifies directly to 13, which is the simplest form.
9. Why is the stepwise division method scoring in CBSE boards?
The stepwise division method is favored in CBSE exams because it:
• Demonstrates a clear and logical approach.
• Matches the prescribed board patterns.
• Provides partial credit even if the final answer is missed.
Hence, students score well by showing methodical steps for √169.
10. When should students round the square root of 169?
Students typically do not need to round the square root of 169 because it is a perfect square with an exact root of 13. Rounding is done when dealing with irrational or non-perfect squares, but since √169 = 13 is exact, rounding is unnecessary in exams.
11. Are there any real-life applications of the square root of 169?
Yes, the square root of 169 (i.e., 13) is used in real-life contexts such as:
• Calculating side lengths of squares with area 169 square units.
• Solving problems in geometry and physics.
• Measuring distances and dimensions where square units apply.
Understanding √169 helps in applying these concepts practically.
12. Is the square root of 169 always an integer?
Yes, since 169 is a perfect square, its square root is always an integer, specifically 13. This differs from numbers that are not perfect squares, whose roots are irrational decimal numbers.
