Answer:
Method: We will use the perfect squares closest to the radicand (the number under the radical symbol) to approximate the square root.
(a) 2√5
Find the perfect squares closest to 5: 4 (less than 5) and 9 (greater than 5).
Since 5 is closer to 4 than 9, we have: 4 < 5 < 9
Take the square root of both sides: 2 < √5 < 3
Since 5 is closer to 4, our approximation is: 2√5 ≈ 2 1/2
Explanation: 2 squared is 4, while 3 squared is 9. Since 5 is closer to 4 than 9, we know its square root must be closer to 2 than 3. Therefore, 2 1/2 is a reasonable approximation for 2√5.
(b) −√8
Find the perfect squares closest to 8: 4 (less than 8) and 9 (greater than 8).
Since 8 is closer to 9 than 4, we have: 9 < 8
Since the square root symbol extracts the positive square root, we know: −√9 < −√8 < 0 (all negative square roots are less than 0)
Take the negative of both sides to obtain positive values for comparison: 0 < √8 < -3 (remember, squaring a negative number results in a positive)
Since 8 is closer to 9, our approximation is: −√8 ≈ -3
Step-by-step explanation:
(c) Plotting the approximations
On the number line, mark 2 1/2 and -3 as shown:
0 1 2 3 4
---------|---------|---------|---------|
-3 2 1/2
Remember, these are just approximations, and the actual values of 2√5 and −√8 lie somewhere between the marked points.
