√14 is between which two integers? X
A. 1 and 2
B. 2 and 3
C. 3 and 4
D. 4 and 5
E. 5 and 7
The Correct Answer and Explanation is :
The correct answer is: C. 3 and 4
To determine between which two integers (\sqrt{14}) lies, we can first find the squares of the integers surrounding it.
- Calculate the squares of nearby integers:
- (3^2 = 9)
- (4^2 = 16)
Since (9 < 14 < 16), it follows that (3 < \sqrt{14} < 4). This means (\sqrt{14}) is between the integers 3 and 4.
- Further verify the approximation of (\sqrt{14}):
We can estimate (\sqrt{14}) by calculating the square roots of numbers that are easier to compute.
- The square root of (13) (which is close to (14)) is approximately (3.605) because (3.6^2 = 12.96).
- The square root of (15) is approximately (3.873) because (3.87^2 = 14.9769).
From these calculations, we see that:
- (\sqrt{14}) is indeed greater than (3.6) and less than (3.873).
- Conclusion:
Therefore, we can confidently state that (\sqrt{14}) falls between the integers (3) and (4). The correct answer to the original question is C. 3 and 4. - General understanding:
Knowing that (\sqrt{n}) can help approximate square roots is beneficial in various mathematical applications. Estimating square roots is often done by identifying perfect squares close to the number in question, allowing for quick approximations without a calculator. This foundational concept in number theory and algebra is essential for higher-level mathematics and real-world applications, such as in physics, engineering, and finance, where such calculations frequently arise.