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 square root of 20?
The square root of 20, denoted as √20, is an irrational number. It cannot be expressed as a simple fraction but can be simplified and approximated. Its approximate value is 4.472.
2. How do you simplify √20?
To simplify √20, use prime factorization: 20 = 2 × 2 × 5 = 2² × 5. Since √(a × a) = a, we can take the square root of 2² as 2, leaving √5 inside the radical. Therefore, the simplified form is 2√5.
3. Is √20 a perfect square?
No, √20 is not a perfect square. A perfect square is a number that can be obtained by squaring an integer. Since there is no integer whose square is 20, it is an imperfect square.
4. What is the value of the square root of 20 up to two decimal places?
The value of √20, rounded to two decimal places, is 4.47.
5. Can you find the square root of 20 without a calculator?
Yes, you can find an approximation without a calculator using methods like the long division method or by estimating based on known perfect squares (e.g., √16 = 4 and √25 = 5, so √20 is between 4 and 5). The prime factorization method gives the simplified form 2√5, requiring only a square root table for 5 to get the approximate decimal value.
6. Is the square root of 20 rational or irrational?
√20 is an irrational number. Irrational numbers cannot be expressed as a fraction of two integers. The decimal representation of √20 is non-terminating and non-repeating.
7. How does √20 compare to √16 and √25?
Since 16 < 20 < 25, we have √16 < √20 < √25, which means 4 < √20 < 5. √20 is between 4 and 5 on the number line.
8. What is the simplest radical form of √20?
The simplest radical form of √20 is 2√5. This is obtained by simplifying the prime factorization of 20.
9. Explain the long division method for finding the square root of 20.
The long division method involves a step-by-step process of finding the square root. It's best explained with a worked example showing the iterative steps to arrive at the approximate decimal value. (Refer to the detailed explanation on the page for the step-by-step process).
10. Are there any real-world applications of the square root of 20?
Yes! The square root of 20 can appear in various geometry problems, such as calculating the diagonal of a rectangle with sides of specific lengths or the radius of a circle given its area. It can also be used in physics and engineering applications involving calculations with magnitudes and distances.
11. How can I use prime factorization to find the square root of 20?
Prime factorization involves breaking down 20 into its prime factors (2 x 2 x 5). Pairing the identical factors (2 x 2), you get 2√5 as the simplified form. This method provides the exact simplest form, although a calculator might be needed to get a decimal approximation from 2√5.
12. What are some common mistakes students make when calculating square roots?
Common mistakes include incorrectly simplifying radicals, not understanding the difference between perfect and imperfect squares, making errors in the long division method, and not properly approximating values. Carefully following the steps and practicing various problems can help avoid these mistakes.
