Evaluate the square root of 21
The Correct Answer and Explanation is:
The square root of 21 is approximately: 21≈4.58257569496\sqrt{21} \approx 4.58257569496
Explanation
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3×3=93 \times 3 = 9.
Now, 21 is not a perfect square—there is no whole number whose square equals 21. The perfect squares closest to 21 are:
- 42=164^2 = 16
- 52=255^2 = 25
This tells us that: 4<21<54 < \sqrt{21} < 5
To find a more accurate value, we use a numerical approximation method, such as:
- Long division method
- Newton-Raphson method
- Using a calculator
With a calculator, we get: 21≈4.58257569496\sqrt{21} \approx 4.58257569496
Irrational Nature
The square root of 21 is an irrational number. That means:
- It cannot be expressed exactly as a fraction.
- Its decimal expansion goes on forever without repeating.
We can write 21≈4.5826\sqrt{21} \approx 4.5826 if we round to 4 decimal places, which is usually sufficient for most practical purposes.
Applications
Understanding square roots is useful in:
- Geometry: For example, calculating the length of a diagonal using the Pythagorean theorem.
- Algebra: Solving quadratic equations.
- Physics and engineering: Where measurements often involve irrational numbers like 2\sqrt{2}, 3\sqrt{3}, or 21\sqrt{21}.
Conclusion
While 21\sqrt{21} cannot be simplified exactly (since 21 has no square factors besides 1), it can be approximated as: 21≈4.5826\boxed{\sqrt{21} \approx 4.5826}
This is the most accurate and practical way to express the square root of 21 in decimal form.
