2F Estimating Square Roots and Order of Operations with Decimals

Let’s look at the distance formula again. The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the Cartesian plane is $$d=\sqrt{[{(x_2-x_1)}^2+{(y_2-y_1)}^2]}$$, but suppose the numbers we are plugging in are decimals. What do we do then? We already went over the order of operations. It will be very similar, but how do we estimate a square root when it is not a perfect square like 4 or 9?

Remember, the square root is the number we can square that gives us the number that is under the square root. For example, $\sqrt{49}$ means the number we can square (multiply by itself) to get 49. The answer is 7, of course, since $7^2=7\times 7=49$. But what happens when the number under the radical is not a perfect square? Perfect squares are 1,4,9,16,25,36,49,64,81,…

Let’s look at $\sqrt{13}$, for example. Can we approximate this radical? What we will do is try to find two numbers that 13 falls in between that we do know the square root of: like 9 and 16. Notice $\sqrt{9}<\sqrt{13}<\sqrt{16}$, but $\sqrt{9}=3$ and $\sqrt{16}=4$ so that implies $3<\sqrt{13}<4$. Hence $\sqrt{13}$ is a decimal between 3 and 4. Notice 13 is in the middle between 9 and 16, but it is slightly closer to 16 than it is to 9. So think of a decimal in the middle of 3 and 4 but slightly closer to 4. Let’s guess 3.6. Remember we are looking for a number that when we square it (multiply times itself), the answer is close to 13. Well, 3.6 x 3.6 = 12.96, which is very close to 13. Hence 3.6 is a pretty good approximation.

Do the following examples with your instructor.

Let’s review order of operations. The order of operations is the order we perform calculations. Remember, if you do a problem out of order, you will often get the wrong answer. Here is the order of operations.

1. Parenthesis $\left(\right)\left[\right]\left\{\right\}$ (Start with the inner-most parenthesis first)
2. Exponents and Square Roots
3. Multiplication and Division in order from left to right.
4. Addition and Subtraction in order from left to right.

For example, let’s look at $0.0013+{(4.7-4.1)}^2÷100$. We do parenthesis first, so 4.7-4.1 = 0.6. Now do the exponents and square roots, so ${0.6}^2=0.6\times 0.6=0.36$ So now we have $0.0013+(0.36)÷100$ Multiplication and division in order from left to right is next. Notice the division must be done before any addition or subtraction. $(0.36)÷100=0.0036$ Adding the last two numbers gives us 0.0013 + 0.0036 = 0.0049 as our final answer.

Now do the following examples with your instructor.

Practice Problems Section 2F

Estimate the following square roots as a decimal that ends in the tenths place. Any approximation that is close is acceptable. Be sure to check your answer.

1. $$\sqrt{3}$$

   Check answer

   1.7 or 1.8

2. $$\sqrt{5}$$

3. $$\sqrt{53}$$

   Check answer

   7.2 or 7.3

4. $$\sqrt{19}$$

5. $$\sqrt{78}$$

   Check answer

   8.8 or 8.9

6. $$\sqrt{41}$$

7. $$\sqrt{23}$$

   Check answer

   4.7 or 4.8

8. $$\sqrt{109}$$

9. $$\sqrt{95}$$

   Check answer

   9.7 or 9.8

10. $$\sqrt{15}$$

11. $$\sqrt{38}$$

    Check answer

    6.1 or 6.2

12. $$\sqrt{22}$$

13. $$\sqrt{97}$$

    Check answer

    9.8 or 9.9

14. $$\sqrt{11}$$

15. $$\sqrt{44}$$

    Check answer

    6.6 or 6.7

16. $$\sqrt{131}$$

17. $$\sqrt{18}$$

    Check answer

    4.2 or 4.3

18. $$\sqrt{76}$$

19. $$\sqrt{24}$$

    Check answer

    4.8 or 4.9

20. $$\sqrt{13}$$

21. $$\sqrt{137}$$

    Check answer

    11.7 or 11.8

22. $$\sqrt{46}$$

23. $$\sqrt{66}$$

    Check answer

    8.1 or 8.2

24. $$\sqrt{10}$$

25. $$\sqrt{82}$$

    Check answer

    9.1

26. $$\sqrt{106}$$

27. $$\sqrt{99}$$

    Check answer

    9.9

28. $$\sqrt{63}$$

29. $$\sqrt{146}$$

    Check answer

    12.1

30. $$\sqrt{6.5}$$

31. $$\sqrt{5.3}$$

    Check answer

    2.3 or 2.4

32. $$\sqrt{165}$$

33. $$\sqrt{8.8}$$

    Check answer

    2.9

Perform the indicated operations for #34-44.

34. $$3.681+10.95÷1.5$$

35. $$15.51-9.26\times 0.68$$

    Check answer

    9.2132

36. $${3.1}^2-3.4\times 0.7$$

37. $$26.3+243.5÷{0.3}^2$$

    Check answer

    $$\mathrm{2,731.8}\overline{5}$$

38. $$2.1+3.6÷3\times 1.5-0.018$$

39. $$44-4.032÷0.315\times 3.3+9.51$$

    Check answer

    11.27

40. $$690.5+1.53÷\sqrt{0.09}\times {10}^3$$

41. $${0.52}^2+\sqrt{0.16}\times 0.77-0.0067$$

    Check answer

    0.5717

42. $$2.6{(1.4+2\times 3.3-7.9)}^3$$

43. $$0.22\times [3.1-{(0.27+0.11)}^2]$$

    Check answer

    0.650232

44. $\sqrt{(20\times 0.36+4.7)}$ (Estimate with a decimal that ends in the tenths place)

Now let’s see if we are ready to tackle the distance formula with decimals. Estimate with a decimal that ends in the tenths place. $$d=\sqrt{[{(x_2-x_1)}^2+{(y_2-y_1)}^2]}$$

45. Find the distance if $$x_1=1.4,x_2=3.8,y_1=7.1,\text{ and }{y}_2=8.2$$

    Check answer

    2.6 or 2.7

46. Find the distance if $$x_1=6.1,x_2=10.2,y_1=1.1,\text{ and }{y}_2=3.3$$

47. Find the distance if $$x_1=2.1,x_2=3.7,y_1=12.4,\text{ and }{y}_2=14.6$$

    Check answer

    2.7 or 2.8

A formula we referenced at the beginning of our book is the certificate of deposit formula $A=P{(1+\frac{r}{n})}^{nt}$.

48. Find the future value A of a certificate of deposit after 1 year (t = 1) if a person invests $2000 (P = $2000) with an annual interest rate of 4% (r = 0.04) compounded twice a year (n = 2).

49. Find the future value A of a certificate of deposit after 2 years (t = 2) if a person invests $5000 (P = $5000) with an annual interest rate of 10% (r = 0.1) compounded once a year (n = 1).

    Check answer

    $6,050

50. Find the future value A of a certificate of deposit after 1 year (t = 1) if a person invests $10000 (P = $10000) with an annual interest rate of 6% (r = 0.06) compounded twice a year (n = 2).

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.