What is the Value of Square Root of 18 in Simplest Radical Form
The concept of square root of 18 is essential in both school-level and competitive mathematics. It appears in algebraic problems, geometry exercises, and real-world applications like measurement and data analysis. Understanding how to find and simplify square roots, including √18, is key for exams and developing strong problem-solving skills.
What is the Square Root of 18?
The square root of 18 refers to the number that, when multiplied by itself, gives 18. The square root symbol is represented as √, so we write it as √18. Since 18 is not a perfect square, its square root is not a whole number, but it can be simplified into a radical (surd) form, and approximated in decimals.
Simplifying Square Root of 18
To simplify √18, follow the steps below:
- Find the prime factors of 18:
18 = 2 × 3 × 3 = 2 × 32 - Pair the prime factors to pull out perfect squares:
√18 = √(2 × 32) - Take out the square root of 32 (which is 3):
√18 = 3√2
So, the simplified square root of 18 is 3√2.
Decimal and Fraction Form of Square Root of 18
| Form | Value |
|---|---|
| Decimal (rounded to 4 places) | 4.2426 |
| Simplest Radical Form | 3√2 |
| Fraction | Cannot be expressed exactly as a fraction (it is an irrational number) |
Since the square root of 18 cannot be written as an exact fraction, it is classified as an irrational number.
Step-by-Step Methods to Calculate √18
1. Prime Factorization Method:
- 18 = 2 × 32
- √18 = √(2 × 32) = 3√2
2. Long Division Method:
- Group digits in pairs: 18.0000 (add pairs of zeros for decimals)
- Find the largest number whose square is less than or equal to 18 (which is 4, since 42=16)
- 4 becomes the first digit. Subtract 16 from 18, bring down next pair of zeros.
- Continue the division for more decimal places. This method gives √18 ≈ 4.2426.
To learn more, visit our detailed guide on How to Simplify Square Roots.
Comparison with Nearby Square Roots
| Number | Square Root | Approx. Value |
|---|---|---|
| 16 | √16 | 4 |
| 18 | √18 | 4.2426 |
| 20 | √20 | 4.4721 |
This shows that √18 is a little more than 4 and less than 4.5.
Worked Examples
Example 1: Simplify √18 + √2
√18 + √2 = 3√2 + √2 = 4√2
Example 2: Find the value of (√18)3
(√18)3 = (3√2)3 = 27 × 2√2 = 54√2
Example 3: If the area of a square is 18 cm2, what is the length of its side?
- Area = side2 → side = √18 = 3√2 cm ≈ 4.24 cm
Practice Problems
- Simplify: √18 ÷ √2
- Is √18 rational or irrational?
- Express √50 in simplest radical form, similar to how you simplified √18.
- Estimate √18 using the interval between √16 and √25.
- If the length of a rectangle is √18 cm and width is 2 cm, find its area.
Common Mistakes to Avoid
- Assuming √18 is a whole number.
- Forgetting to pair prime factors when simplifying.
- Confusing √18 with 18 ÷ 2 or 9 (incorrect calculation methods).
- Writing √18 as a simple fraction (it is an irrational number).
Real-World Applications
Square roots like √18 come up in geometry (e.g., finding side lengths of squares or rectangles with known area), in science for calculations involving square laws, and in engineering when working with surds or irrational numbers. For example, if a chessboard has side length 3√2 cm, its area is precisely 18 cm2.
Understanding how to work with square roots is also critical for number system questions in competitive exams.
In summary, the square root of 18 is an important example for learning to simplify radicals, approximate decimals, and identify irrational numbers. At Vedantu, we help students master these foundational concepts step-by-step. Remember: √18 = 3√2 ≈ 4.2426, and practicing square roots improves your mathematical skills for school and beyond.
FAQs on Square Root of 18 Explained with Simplification
1. What is the square root of 18?
The square root of 18 is √18 = 3√2 ≈ 4.2426. Since 18 is not a perfect square, its square root is irrational. It can be simplified by factoring 18 as 9 × 2, where √9 = 3. Therefore, √18 = √(9 × 2) = 3√2, which in decimal form is approximately 4.2426.
2. How do you simplify the square root of 18?
To simplify √18, write it as 3√2 by factoring out the perfect square. Follow these steps:
- Factor 18 as 9 × 2.
- Take the square root of 9, which is 3.
- Leave √2 inside the radical.
3. Is the square root of 18 a rational or irrational number?
The square root of 18 is an irrational number. An irrational number cannot be written as a simple fraction and has a non-terminating, non-repeating decimal. Since 18 is not a perfect square, √18 ≈ 4.242640687... continues infinitely without repeating.
4. What is the decimal value of the square root of 18?
The decimal value of √18 is approximately 4.2426. Using a calculator, √18 ≈ 4.242640687..., which is a non-terminating and non-repeating decimal because it is irrational.
5. How do you find the square root of 18 using prime factorization?
You can find √18 using prime factorization and simplify it to 3√2. Steps:
- Prime factorize 18 = 2 × 3 × 3.
- Group the pair of 3s: (3 × 3).
- Take one 3 outside the radical.
6. What is the value of the square root of 18 in simplest radical form?
The simplest radical form of √18 is 3√2. This form is obtained by factoring out the largest perfect square (9) from 18 and simplifying the radical expression.
7. Between which two integers does the square root of 18 lie?
The square root of 18 lies between 4 and 5. Since 4² = 16 and 5² = 25, and 18 is between 16 and 25, √18 must be between 4 and 5. Its approximate value is 4.2426.
8. What is the square of √18?
The square of √18 is 18. Squaring a square root cancels the radical: (√18)² = 18. Similarly, (3√2)² = 9 × 2 = 18.
9. Can the square root of 18 be written as a fraction?
The square root of 18 cannot be written as an exact fraction because it is irrational. While you can approximate it as 42426/10000 or 4.2426, these are only rounded decimal approximations, not exact fractional forms.
10. How is the square root of 18 used in real-life or geometry problems?
The square root of 18 is commonly used in geometry and the Pythagorean theorem to find lengths of sides in right triangles. For example:
- If a right triangle has legs of 3 units and 3 units, then the hypotenuse = √(3² + 3²) = √18.
- This simplifies to 3√2.
