1D Formulas with Square Roots and Order of Operations

A common formula in algebra is the distance between two points. $$d=\sqrt{[{(x_2-x_1)}^2+{(y_2-y_1)}^2]}$$

This formula gives many students problems as it involves different operations that must be done in a particular order. The different letters and parentheses can be confusing as well. We see that to make sense of a formula like this, we need to not only review square roots, but also review the order of operations.

A square root is a number we can square that gives us the number that is under the square root. For example, $\sqrt{49}$ means what number can we square (multiply by itself) to get 49? The answer is 7 of course, since $7^2=7\times 7=49$.

Do the following examples with your instructor.

The order of operations is the order we perform calculations. If you do a problem out of order, you will often get the wrong answer. Here is the order of operations. It is vital to memorize these!

Order of Operations

1. Parentheses $\left(\right)\left[\right]\left\{\right\}$
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 $89-(13+17)÷6\times 2^3$. We do parenthesis first, so 13 + 17 is 30. We now do exponents, so $2^3=2\times 2\times 2$=8. So now we have 89 – (30)$÷$ 6 x 8. Multiplication and division in order from left to right is next. Notice a division came first so we do the division before the multiplication. 30 divided by 6 is 5. Now 5 x 8 = 40. So we are left with 89-40 = 49.

Now do the following examples with your instructor. Pay close attention to doing the problems in the correct order.

Practice Problems Section 1D

Perform the indicated operation for #1-35.

1. $$\sqrt{1}$$

   Check answer

   1

2. $$\sqrt{0}$$

3. $$\sqrt{16}$$

   Check answer

   4

4. $$\sqrt{4}$$

5. $$\sqrt{49}$$

   Check answer

   7

6. $$\sqrt{64}$$

7. $$\sqrt{25}$$

   Check answer

   5

8. $$\sqrt{9}$$

9. $$\sqrt{100}$$

   Check answer

   10

10. $$\sqrt{81}$$

11. $$\sqrt{121}$$

    Check answer

    11

12. $$\sqrt{196}$$

13. $$\sqrt{400}$$

    Check answer

    20

14. $$\sqrt{225}$$

15. $$\sqrt{144}$$

    Check answer

    12

16. $$\sqrt{8100}$$

17. $$\sqrt{900}$$

    Check answer

    30

18. $$\sqrt{169}$$

19. $$\sqrt{2500}$$

    Check answer

    50

20. $$\sqrt{1600}$$

21. $$\sqrt{4900}$$

    Check answer

    70

22. $$71-3\times 13$$

23. $$48+60÷12$$

    Check answer

    53

24. $$99-24÷6\times 11$$

25. $$6\times \sqrt{4}+7^2$$

    Check answer

    61

26. $$21+36÷3\times 5-18$$

27. $$69-5\times \sqrt{9}+{10}^2$$

    Check answer

    154

28. $$48-36\times 4÷3^2$$

29. $$79-15+4^2\times \sqrt{81}$$

    Check answer

    208

30. $$7{(14+2\times 3-19)}^3$$

31. $$56+4\times 18÷9-{(3+\sqrt{4})}^2$$

    Check answer

    39

32. $$55+(23-17)\times 6+4^3$$

33. $$54÷\sqrt{81}+{(13-7)}^2$$

    Check answer

    42

34. $$4[44÷(5-1)\times \sqrt{100}]$$

35. $$\frac{[4\times {(18-13)}^2]}{\sqrt{(15-11)}}$$

    Check answer

    50

Remember the z-score formula in statistics. Here is a tougher z-score formula that many statistics students struggle with. Let’s see if you can calculate it. Remember to use the order of operations. $z=\frac{(\overline{x}-\mu )}{(\sigma ÷\sqrt{n})}$

36. Find the z-score when $\overline{x}=51$, $\mu =46$, $\sigma =6$ and $n=9$.

37. Find the z-score when $\overline{x}=25$, $\mu =17$, $\sigma =32$ and $n=16$.

    Check answer

    1

38. Find the z-score when $\overline{x}=139$, $\mu =121$, $\sigma =12$ and $n=4$.

39. Find the z-score when $\overline{x}=2596$, $\mu =2500$, $\sigma =120$ and $n=25$.

    Check answer

    4

Now let’s see if we are ready to tackle the distance formula: $$d=\sqrt{[{(x_2-x_1)}^2+{(y_2-y_1)}^2]}$$

40. Find the distance if $$x_1=5,x_2=13,y_1=10,\text{ and }{y}_2=16$$

41. Find the distance if $$x_1=6,x_2=10,y_1=11,\text{ and }{y}_2=14$$

    Check answer

    5

42. Find the distance if $$x_1=3,x_2=8,y_1=2,\text{ and }{y}_2=14$$

43. Find the distance if $$x_1=17,x_2=26,y_1=13,\text{ and }{y}_2=25$$

    Check answer

    15

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