How to Find the Square Root of 169 Using Prime Factorization Method
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 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 What Is the Square Root of 169
1. What is the square root of 169?
The square root of 169 is 13. This is because 13 × 13 = 169, so when 169 is multiplied by itself through 13, it gives the original number. In mathematics, the square root of a number is the value that, when multiplied by itself, equals the given number.
2. Is the square root of 169 positive or negative?
The principal square root of 169 is +13, but the number actually has two square roots: +13 and −13. By definition, the square root symbol (√) represents only the positive value, called the principal square root. However, both 13 × 13 and −13 × −13 equal 169.
3. How do you find the square root of 169?
You can find the square root of 169 by identifying a number that multiplies by itself to give 169, which is 13.
- Check perfect squares: 10² = 100, 11² = 121, 12² = 144, 13² = 169.
- Since 13² = 169, the square root is 13.
4. Is 169 a perfect square?
Yes, 169 is a perfect square because it is the square of a whole number, 13. A perfect square is any number that can be written as n², where n is an integer. Since 13 × 13 = 169, it satisfies this definition.
5. What is the prime factorization of 169?
The prime factorization of 169 is 13 × 13 or 13². Since 13 is a prime number and 169 equals 13 multiplied by itself, it has only one prime factor repeated twice. This also confirms that the square root of 169 is 13.
6. What is the value of √169 using the long division method?
Using the long division method, the value of √169 is 13.
- Pair the digits: 1 | 69.
- Find the largest number whose square is ≤ 1 → 1² = 1.
- Bring down 69 to make 69.
- Double 1 to get 2 and find a digit that makes 2_ × _ ≤ 69.
- Choosing 3 gives 26 × 3 = 78 (too big), but 23 × 3 = 69 works correctly.
7. What is the difference between √169 and −√169?
The value of √169 is +13, while −√169 equals −13. The square root symbol (√) gives the principal positive root, but adding a negative sign outside the radical changes the value to negative. Both numbers square to give 169.
8. Is the square root of 169 a rational number?
Yes, the square root of 169 is a rational number because it equals 13, which is an integer. Rational numbers can be written in the form p/q, and since 13 can be written as 13/1, it satisfies the definition of a rational number.
9. How do you verify that 13 is the square root of 169?
You verify that 13 is the square root of 169 by squaring it to check if it equals 169.
- Calculate: 13 × 13.
- The result is 169.
10. What are the properties of the square root of 169?
The square root of 169 has key properties: its principal value is 13, and it is a perfect square root.
- It is a whole number and an integer.
- It is a rational number.
- It has two roots: +13 and −13.
- It comes from the prime factorization 13².
