4E Exponents and Order of Operations with Negative Quantities

A common formula in Chemistry is the formula to convert degrees Fahrenheit into degrees Celsius. $C=\frac{(F-32)}{1.8}$. Since temperature often involves negative quantities, we see that we need to make sure that we can use the order of operations correctly.

The main key is to remember that the order of operations does not change just because some or all of the numbers are negative. We still do parenthesis first, then exponents, then multiplication or division in order from left to right, and last addition and subtraction from left to right.

Exponents can be particularly tricky with negative numbers. We need to make sure that we keep in mind whether or not the number is in parenthesis or not. For example:

For ${(-7)}^2$, notice that the -7 is in parenthesis, so the negative is applied first, then the square. Hence ${(-7)}^2=-7\times -7=+49$.

But what happens when there is no parenthesis?

If we see $-7^2$, this is saying subtract 7 squared. By order of operations we do the square first, then subtract. Hence we see that $7^2=7\times 7=49$, then we subtract. Hence $-7^2=-49$

Try the following problems with your instructor.

Practice Problems Section 4E

Simplify the following.

1. $$-9^2$$

   Check answer

   ^-81

2. $$-2^4$$

   Check answer

   ^-16

3. $${(-12)}^2$$

   Check answer

   144

4. $${(-2)}^4$$

   Check answer

   16

5. $${(-6)}^3$$

   Check answer

   ^-216

6. $$-{20}^2$$

   Check answer

   -400

7. $$-2^3$$

   Check answer

   ^-8

8. $${(-10)}^2$$

   Check answer

   100

9. $$-8^2$$

   Check answer

   ^-64

10. $$-1^5$$

    Check answer

    ^-1

11. $${(-7)}^3$$

    Check answer

    -343

12. $${(-13)}^2$$

    Check answer

    169

13. $${(-3)}^4$$

    Check answer

    81

14. $$-4^3$$

    Check answer

    ^-64

15. $${(-5)}^3$$

    Check answer

    ^-125

16. $${(-10)}^4$$

    Check answer

    10,000

17. $$-{15}^2$$

    Check answer

    ^-225

18. $$-{(-6)}^2$$

    Check answer

    ^-36

19. $$-7+-4\times -3$$

    Check answer

    5

20. $${(-2)}^4-8^2$$

    Check answer

    ^-48

21. $$(-12-+9)÷(-7)$$

    Check answer

    3

22. $${(-15)}^2-4^2$$

    Check answer

    209

23. $$-7(+8)÷(-2\times +4)+(-12)$$

    Check answer

    ^-5

24. $${(-8--7)}^3+-24÷+3$$

    Check answer

    -9

25. $$-6[{(+13+-14)}^3\times (-7)]$$

    Check answer

    ^-42

26. $${(-13+-7)}^2÷(-100)$$

    Check answer

    ^-4

Use the chemistry formula $F=1.8\times C+32$ to convert Celsius into Fahrenheit.

27. Convert $-4\text{°}C$ into degrees Fahrenheit.

    Check answer

    24.8 $\text{°F}$

28. Convert $-13\text{°}C$ into degrees Fahrenheit.

    Check answer

    8.6 $\text{°F}$

Use the chemistry formula $C=\frac{(F-32)}{1.8}$ to convert Fahrenheit into Celsius.

29. Convert $26.6\text{°}F$ into degrees Celsius.

    Check answer

    ^-3 $\text{°C}$

30. Convert $-2.2\text{°}F$ into degrees Celsius.

    Check answer

    ^-19 $\text{°C}$

(For #31-33) In statistics, we use regression formulas to help businesses make predictions about things like profits and costs. A company makes flat screen TVs. We used regression to find the formula $$P=-12.8{(x-43)}^2+\mathrm{19,780}$$ where x represents the number of TVs made per day and P represents the profit in dollars.

31. If a company makes 23 TVs in a day, how much profit will they expect to make?

    Check answer

    $14,660

32. If the equipment malfunctions and the company cannot make any TVs on one day, how much money will we expect them to lose?

    Check answer

    Lose $3887.20

33. We calculated that the company should strive to make 43 TVs per day, as that value maximizes their profit. What would their maximum profit be if they made exactly 43 TVs in a day?

    Check answer

    $19780

34. A small business starts out with a net loss of $-\mathrm{1,200}$ dollars per month over the first 4 months in operation, and follows that with a net profit of $+800$dollars per month over the next 8 months. How much money did the business make or lose in its first year?

    Check answer

    Made $1,600

35. A football running back loses 4 yards per carry for 5 of his rushing attempts. He also gains 8 yards per carry for 2 of his attempts. What are his total yards gained or lost?

    Check answer

    ^-4 yards

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