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Square Root of 18: How to Find, Simplify, and Apply

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Step-by-Step Guide: Simplifying and Calculating √18

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:

  1. Find the prime factors of 18:
    18 = 2 × 3 × 3 = 2 × 32
  2. Pair the prime factors to pull out perfect squares:
    √18 = √(2 × 32)
  3. 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:

  1. 18 = 2 × 32
  2. √18 = √(2 × 32) = 3√2

2. Long Division Method:

  1. Group digits in pairs: 18.0000 (add pairs of zeros for decimals)
  2. Find the largest number whose square is less than or equal to 18 (which is 4, since 42=16)
  3. 4 becomes the first digit. Subtract 16 from 18, bring down next pair of zeros.
  4. 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?

  1. 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: How to Find, Simplify, and Apply

1. What is the square root of 18?

The square root of 18, denoted as √18, is the number that when multiplied by itself equals 18. It's an irrational number, meaning it cannot be expressed as a simple fraction.

2. What is the simplest form of √18?

The simplest radical form of √18 is 3√2. This is achieved by finding the prime factorization of 18 (2 x 3 x 3) and simplifying the square root.

3. How do you simplify the square root of 18?

To simplify √18:

  • Find the prime factorization of 18: 2 x 3 x 3 = 2 x 32
  • Rewrite the square root: √(2 x 32)
  • Separate the perfect square: √(32) x √2
  • Simplify: 3√2

4. What is the value of √18 in decimal?

The decimal approximation of √18 is approximately 4.2426. Remember that this is an approximation because √18 is an irrational number.

5. Is the square root of 18 a rational number?

No, the square root of 18 is an irrational number. Irrational numbers cannot be expressed as a fraction of two integers.

6. Can √18 be expressed as a fraction?

No, √18 cannot be expressed as a simple fraction because it is an irrational number. Its decimal representation goes on forever without repeating.

7. Why is the square root of 18 written as 3√2?

√18 is written as 3√2 because it's the simplified radical form. Simplifying radicals means expressing them in their most reduced form, removing any perfect square factors from under the radical sign.

8. How does √18 compare to other square roots like √16 and √25?

√18 falls between √16 (which is 4) and √25 (which is 5). This is because 16 < 18 < 25. Understanding this relationship helps with estimations.

9. What is the geometric interpretation of √18?

Geometrically, √18 represents the length of the side of a square with an area of 18 square units. It also relates to the hypotenuse of a right-angled triangle with certain side lengths.

10. How to estimate √18 without a calculator?

Since 4² = 16 and 5² = 25, √18 is between 4 and 5. Because 18 is closer to 16 than 25, a reasonable estimate would be slightly above 4. More precise estimation methods exist, but this is a good starting point.

11. What is the significance of simplifying roots in algebraic expressions?

Simplifying radicals in algebra simplifies calculations and makes it easier to work with expressions. For example, in solving equations or simplifying larger expressions. It also helps to see relationships and patterns more clearly.