How to Simplify the Square Root of 20 Manually?
The concept of square root of 20 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Square Root of 20?
A square root of 20 is a number that, when multiplied by itself, yields 20. The square root is represented with the √ symbol, i.e., √20. You’ll find this concept applied in areas such as algebra, geometry, and word problems in Mathematics.
Key Formula for Square Root of 20
Here’s the standard formula: \( \sqrt{20} = 2\sqrt{5} \) (in simplest radical form)
Cross-Disciplinary Usage
The square root of 20 is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in distance, area, and measurement questions.
Is 20 a Perfect Square?
No, 20 is not a perfect square. A perfect square has integer square roots (like 4, 9, 16), but √20 is not an integer. This means its square root value will always be a decimal or in radical form.
How to Find the Value of Square Root of 20?
The value of the square root of 20 can be found by estimation, division method, or using the prime factorization method:
| Form | Answer |
|---|---|
| Radical Form | √20 |
| Simplest Form | 2√5 |
| Decimal (approx.) | 4.472 |
Step-by-Step Illustration
- Simplify using Prime Factorization:
1. Write 20 as product of prime factors: 20 = 2 × 2 × 5
2. Pair the same numbers: (2 × 2) forms a pair; 5 remains
3. Take one number from the pair outside the root: 2
4. Final Answer: √20 = 2√5
- Estimate by Nearest Perfect Squares:
1. √16 = 4 and √25 = 5
2. So, √20 is between 4 and 5.
3. Calculate (optional): √20 ≈ 4.472
- By Long Division:
1. Place number 20 under bars (20.0000...)
2. Find a number whose square is ≤ 20 (4 × 4 = 16)
3. Place 4 in quotient; Subtract 16 from 20 (remainder 4); bring down next pair of zeros
4. Double the quotient (8), use as new divisor, find next digit, continue.
5. After a few steps: √20 ≈ 4.472
Speed Trick or Vedic Shortcut
Here’s a quick shortcut to simplify √20 during exams:
- Break 20 into multiples: 20 = 4 × 5
- Write √20 = √4 × √5 = 2√5
For other roots, apply similar logic! See Vedantu's square root tricks for more tips.
Is Square Root of 20 Rational or Irrational?
The square root of 20 is an irrational number. Its decimal expansion never terminates or repeats (4.472135954...). It cannot be written as an exact fraction. This is true for all non-perfect squares.
Visual Representation of √20
To see where √20 lies, plot it on the number line between 4 and 5. Use number line methods for practice.
Uses and Word Problems with √20
The square root of 20 is seen in real-life geometry, for example:
- The diagonal of a rectangle with sides 2 units and √20 units
- Finding the side of a square when area is 20 sq units
Try problems with area of squares or Pythagoras calculations where the answer comes as √20!
Try These Yourself
- Express √20 in its simplest radical form.
- Calculate the value of 2 × √20.
- Show why √20 is irrational.
- Compare √20 to √16, √25, and √10.
Frequent Errors and Misunderstandings
- Assuming √20 = 4 or 5 (it is a decimal, not an integer).
- Forgetting to simplify to 2√5 in board exams.
- Using rounded approximations (4.47) when exact forms (2√5) are needed.
Relation to Other Concepts
The square root of 20 connects closely with perfect squares, other imperfect roots, and prime factorization in algebra. Understanding this makes roots easier in algebraic expressions and geometry.
Classroom Tip
A quick way to remember √20’s simplest form: if the number inside has a perfect square, split it! For example, √20 = √4 × √5 = 2√5. Vedantu’s teachers often use such cues in live doubt-solving classes.
Related Roots and Comparison Table
| Root | Simplest Form | Decimal |
|---|---|---|
| √16 | 4 | 4.000 |
| √20 | 2√5 | 4.472 |
| √25 | 5 | 5.000 |
| √10 | (Cannot be simplified) | 3.162 |
We explored square root of 20—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu’s theory on squares and square roots and use the square root calculator to master this concept!
Explore More: Square Root of 4 | Square Root of 10 | Prime Factorization for Roots | Estimating Square Root
FAQs on Square Root of 20 – Steps, Value, Simplified Form & More
1. What is the value of the square root of 20?
The square root of 20, represented as √20, is an irrational number. Its value cannot be expressed as a simple fraction. The approximate decimal value of √20 is 4.472. When rounded to two decimal places, it is 4.47.
2. What is the square root of 20 in its simplified radical form?
To find the simplified radical form, we use prime factorization. The prime factors of 20 are 2 × 2 × 5, or 2² × 5. By taking the square root, we can pull the pair of 2s out of the radical. Therefore, the simplified form of √20 is 2√5.
3. How is the square root of 20 found using the long division method?
The long division method is a step-by-step process to find an accurate decimal value for an imperfect square like 20.
Start by pairing the digits. For 20, we can write it as 20.00 00.
Find the largest number whose square is less than or equal to 20 (which is 4, since 4² = 16).
Subtract 16 from 20 to get a remainder of 4. Bring down the next pair of zeros.
Double the quotient (4) to get 8, and find a digit 'x' such that 8x × x is close to 400. The digit is 4 (84 × 4 = 336).
Continue this process to get more decimal places. This method gives the precise value of 4.472...
4. Is the square root of 20 a rational or irrational number?
The square root of 20 is an irrational number. This is because 20 is not a perfect square (no integer multiplied by itself equals 20). As a result, its decimal representation is non-terminating and non-repeating, which is the definition of an irrational number.
5. What is the difference between the 'square of 20' and the 'square root of 20'?
These are two opposite mathematical operations.
The square of 20 means multiplying 20 by itself (20 × 20), which equals 400.
The square root of 20 (√20) is the number that, when multiplied by itself, equals 20. This value is an irrational number, approximately 4.472.
6. How can you estimate the value of √20 without a calculator?
You can estimate the value of √20 by identifying the two closest perfect squares. We know that √16 = 4 and √25 = 5. Since 20 lies between 16 and 25, the value of √20 must lie between 4 and 5. Because 20 is closer to 16 than to 25, we can estimate that its square root will be closer to 4, which aligns with its actual value of approximately 4.472.
7. What are some real-world examples where the square root of 20 might be used?
The square root of 20 can appear in various practical scenarios, especially in geometry and physics. For example:
Geometry: Calculating the length of the hypotenuse of a right-angled triangle with sides of length 2 and 4 units (c² = 2² + 4² = 4 + 16 = 20, so c = √20).
Area Calculation: Finding the side length of a square-shaped garden that has an area of 20 square metres.
Physics: Problems involving vector magnitudes or distances in a coordinate plane might result in calculations requiring √20.
8. Why is simplifying √20 to 2√5 considered a better mathematical practice?
Simplifying √20 to 2√5 is preferred because it expresses the number in its most reduced radical form. This has several advantages:
Clarity: It separates the rational part (2) from the irrational part (√5), making the number easier to understand and compare.
Calculation: It simplifies further algebraic operations. For instance, adding √20 + √5 is easier when written as 2√5 + √5 = 3√5.
Standard Form: It is the conventional way to write radicals, ensuring consistency in mathematical notation.
9. What are some common mistakes to avoid when working with the square root of 20?
A common mistake is incorrectly simplifying the radical. For example, students might mistakenly write √20 as 4√5 by taking the square root of a factor instead of finding pairs of prime factors. Another error is confusing √20 with 20/2; the square root is approximately 4.472, while half of 20 is 10. It is also important not to treat it as a rational number and prematurely round it in multi-step calculations.
10. If a square has an area of 20 square units, what is the exact length of its side?
The area of a square is calculated by the formula Area = side × side (or side²). If the area is 20 square units, then side² = 20. To find the length of one side, we must take the square root of the area. Therefore, the exact length of the side is √20 units, which simplifies to 2√5 units.
