7.5 Estimating Values of Expressions
Surprise! We’re going to use rational approximations of irrational numbers again. We’re basically going to
be looking at adding, subtracting, and multiplying irrational numbers and how to quickly estimate an answer.
For example, let’s try to add the following irrational numbers: √37 . √24. As a very fast estimate, we
know that √37 ≈ 6 since 37 is just over 36 and √24 ≈ 5 since 24 is just under 25. That means we would estimate
√37 . √24 ≈ 11.
We could further fine tune our estimates by approximating the irrational numbers to one decimal place.
Here’s an example:
√37 . √56 U 6.1 + 7.5 U 13.6
One last concept we need to be familiar with is multiplication involving irrational numbers. Recall that the
expression 5 means five times . In the same way, 5√15 means five times the square root of fifteen. To estimate
that expression we can approximate in the following way:
5√15 U 5(4) U 20 or with more precision 5√15 U 5(3.9 ≈ 19.5
Now combining all of those qualities we can estimate more complex expressions involving all of the
operations of addition, subtraction, and multiplication. Just don’t forget to follow the order of operations! For
example, to the nearest whole number we could estimate the following expression:
2√13 + 5√5 − √37 ≈ 23.5) + 5(2) − 6 U 11
Note that it was useful to approximate √13 to 3.5 in the middle of the problem since it led to a nice whole
number solution at the end. In general, if we want a whole number answer, it might be a good idea to approximate
each irrational as either a whole number or the nearest half value.
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Lesson 7.5
Estimate the following expressions to the nearest whole number.
1. √8 + √18
2. 11 − √80
3. 4√48
4. 3√24 . 3
5. 2√35 − 3√8
6. √14 + √26
7. √120 − 7
8. 2√63
9. 4√15 − 5
10. 2√66 − 3√5
11. √9 . √10
12. 20 − √102
13. 2√15
14. 3√15 + 1
15. 4√24 − 3√3
16. √14 + √34
17. √105 − 9
18. 5√26
19. 2√83 − 8
20. 3√17 − 2√1
21. √47 + √8
22. 8 − √48
23. 7√10
24. 4√5 + 9
25. 3√24 − 5√5
26. √65 + √63
27. √100 − 2
28. 6√5
29. 2√26 − 3
30. 4√26 − 3√4
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