Find two consecutive whole numbers that \sqrt{89} lies between
The Correct Answer and Explanation is :
The correct answer is:9 and 10
To find two consecutive whole numbers between which (\sqrt{89}) lies, we first need to identify the whole numbers that are perfect squares close to 89.
- Identifying Perfect Squares:
- The square of 9 is (9^2 = 81).
- The square of 10 is (10^2 = 100). Since (81 < 89 < 100), we can conclude that (\sqrt{89}) lies between the square roots of these two perfect squares.
- Finding the Consecutive Whole Numbers:
- Therefore, (\sqrt{89}) is between 9 and 10.
- Estimating (\sqrt{89}):
- To confirm this, we can estimate (\sqrt{89}) more closely.
- Calculating the value gives approximately:
[
\sqrt{89} \approx 9.434
] This value is indeed greater than 9 but less than 10, which reinforces our previous conclusion.
- Conclusion:
Thus, the two consecutive whole numbers that (\sqrt{89}) lies between are 9 and 10.
Explanation:
Understanding square roots is fundamental in mathematics. The square root of a number (x) is a value (y) such that (y^2 = x). Whole numbers that result from squaring integers are known as perfect squares. For example, the perfect squares of 1 to 10 are (1, 4, 9, 16, 25, 36, 49, 64, 81, 100).
When estimating square roots, knowing the closest perfect squares helps narrow down the interval in which the square root lies. By identifying the perfect squares surrounding 89, we determined that (\sqrt{89}) falls between the squares of 9 and 10, confirming it lies between these two consecutive integers.
Using the technique of estimation provides not just the interval, but a more intuitive understanding of square roots and their values, which is especially useful in various mathematical applications, including algebra, geometry, and real-world problem-solving.