Lesson 7 · from §16 of the 1910 text
Putting It Together
This lesson mixes all four operations with signed numbers. Two writing conventions first:
- If the first of several numbers joined by + or - signs is positive, its sign is usually left off: +4 - 3 + 6 is written 4 - 3 + 6. A negative first number always keeps its sign.
- When two numbers in parentheses sit side by side with no sign between them, they're multiplied: (6)(3), 6(3), and (6)3 all mean 6 \cdot 3.
Quick check
(-4)(5) - 3
Powers of negative numbers follow from the multiplication law. An even number of negative factors gives a positive product, and an odd number gives a negative one: (-2)^2 = +4, \qquad (-2)^3 = -8, \qquad (-2)^4 = +16 Careful: (-3)^2 = 9, because the parentheses say to square -3.
Quick check
(-1)^7
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Simplify. (A first number with no sign is positive, and numbers in side-by-side parentheses are multiplied.)
(7)+(5)
(7)-(5)
(7)+(-5)
(7)-(-5)
(-7)+(5)
-7+5
(-9)-(4)
-9-4
-11+(-13)
-6-(-10)
-6+10
8+(-10)
12-18
-18-12
15-14
+7-0
-0-3
(-3)(6)
(-5)6
(7)(-5)
8(-3)
(-5)(-12)
-3(-8)
-5 \cdot 4
5 \cdot 0
0 \cdot (-9)
4 \cdot 8
-3 \cdot 6
12 \div (-2)
-12 \div 2
-39 \div (-3)
45 \div (-15)
0 \div (-6)
0 \div 3
-27 \div 9
3-5+6
-4+6-2+1
-4+6+2-1
2-3+4-5-6
Add each column.
Add: \begin{array}{r}7\\ -2\\ 3\\ \underline{\;-5\;}\end{array}
Add: \begin{array}{r}6\\ -2\\ -3\\ \underline{\;4\;}\end{array}
Add: \begin{array}{r}-8\\ 6\\ 2\\ \underline{\;-5\;}\end{array}
Add: \begin{array}{r}4\\ -9\\ -3\\ \underline{\;6\;}\end{array}
Simplify.
3 \cdot 6 \div 3
-4(7) \div (-2)
3(-6) \div 2
4 \cdot 6(-8) \div (-16)
18 \div (-3) \cdot 6 \div 4
3^2
(-3)^2
2^3
(-2)^3
(-4)^3 + (4)^2
(-1)^2 + (-1)^3 + (-2)^2 + (-2)^3
6 + 3 \cdot 2 + 18 \div (-3)
5^2 - 4 \div (-2) + 6(-3)
3 \cdot 6 \div 9 - 2 \cdot 6 \div 4 + (-3)^2
6 + 6 \cdot 3^2 - 5^2 \cdot 2 \div 10
If x = 3 and y = -2, find the value of each expression.
y^2
y^3
y^4
y^5
2y^2
2y^3
5x^2y^2
4x^2y^4
x^2 - y^2
x^3 - y^3
x^2 + 2xy + y^2
x^2 - 2xy + y^2
(x + y)(x - y)
x^3 + 3x^2y + 3xy^2 + y^3
y^3 - 3xy^2 + 3x^2y - x^3
Check whether each equation is true for the given values.
Does 4x - 2 = 2x + 8 if x = 5?
Does 3x - 5 = 2x + 8 if x = -9?
Does x^2 - x - 12 = 0 if x = 4? if x = -8? if x = -4?
Does 3x^2 + 19x = 14 if x = \tfrac{2}{3}? if x = 2? if x = -7?
Problems
At 7:00 A.M. one day the thermometer read 15 degrees above zero. The mercury then fell 3 degrees per hour. What was the temperature at 9:00 A.M.? at noon? at 3:00 P.M.?
Measured from a certain level, a surveyor found the heights of 5 points to be +30 ft, -7 ft, +18 ft, -10 ft, and +16 ft. What was the average height of the 5 points? (A positive answer means above the level.)
A sixth point, -38 feet high, was later added to the survey in Exercise 79. Find the mean height of the 6 points.
Later a seventh point was added. The average height of all 7 points was then zero. Find the height of the seventh point.
Euclid lived about 300 B.C. Sir Isaac Newton died in 1727 A.D. If dates before Christ are negative and dates after are positive, how might these dates be written?
The date -450 means 450 B.C., and +1910 means 1910 A.D. How many years apart are these two dates?
In still water a gasoline launch travels 8 miles per hour. Represent its rate up and down a river that runs 1\tfrac{1}{2} miles per hour.
A boat travels 12 miles per hour. A man on its deck walks 3 miles per hour. Represent the rate at which he approaches his destination when he walks toward the bow and when he walks toward the stern.
A balloon that can lift 500 pounds is held down by 10 men averaging 150 pounds each. Using signed numbers (lift as negative weight), represent the weight of the balloon, of the men, and of the balloon and men together.
A man swims in still water at 1\tfrac{1}{2} miles per hour. In a river flowing 2 miles per hour, represent his rate (a) swimming downstream and (b) swimming upstream. (The negative answer means the river carries him backward.)