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13. Do you know of any game in which there are negative counts ? Describe the method of counting. How do you find the total score?

14. Express as a continued addition the bank account of a man who (1) has $67.50, (2) spends $5.46, (3) deposits $21.00, (4) spends $41.25, (5) spends $12.50, (6) deposits $14.15, (7) spends $7.20. What will his final bank account be?

23. Multiplication of Positive and Negative Numbers. · In arithmetic the numbers which we multiply together are always positive, but in algebra some (or all) of the numbers to be multiplied may be negative. Four different cases are here possible and will now be illustrated.

(1) Consider the product 5X4. This means, as in arithmetic, that 5 is to be taken 4 times. Hence the result is 20; that is,

5X4=20. (2) Consider the product (-5) X4. This means that -5 is to be taken 4 times. Hence, by the Rule in $ 17, the result must be – 20; that is,

(-5) X4= –20. (3) Consider the product 5X(-4). Remembering that in arithmetic we have 2x3=3X2, we may here suppose that 5X(-4)=(-4) X5. But this last form (being similar to the one in the second case) means – 4 taken 5 times, which is – 20. Thus, we have

5X(-4)= –20. (4) Consider the product (-5)X(-4). We naturally look upon this as meaning the negative of 5X(-4); that is, the negative of — 20. But the negative (or opposite) of – 20 is +20, or simply 20. Whence, we write

(-5)X(-4)=20.

These four illustrations make clear the following rule.

RULE FOR MULTIPLICATION OF Two NUMBERS. Multiply as in arithmetic, prefixing the + sign to the product if the two numbers have the same sign, and prefixing the sign to the product if the two numbers have opposite signs.

9. (-a)X(-b). 17. (-1)^(-2)

-40). 21. (-1)(-8).

ORAL EXERCISES State the value of each of the following expressions : 1. 3X2.

10. (-) x3 17. (-1)XI .. 2. 3X(-2). 3. (-3)X2. 11. (-i)X1. 18.(-2+1)X(3–4). 4. (-3) X(-2). 12. (-})XÅ 19. X(-). 5. 4X(-7). 13. (-{)X(-1). 6. (-10) X(-6). 14. (-s)X(-). 7. aX(-6). 15. 3X(-x2). 8. (-a) Xb. 16. (-x2) x(-y).

24. Multiplication of Several Numbers. The product of three or more numbers is found by performing one multiplication at a time. Thus, in finding 6 X(-5) X(-4) X3 the steps are as follows:

6X(-5)= -30, (-30)X(-4)=120,

120X3=360. Therefore, 6X(-5) X(-4) X3=360. Ans.

NOTE 1. If the number of negative factors in a product is even, the sign of the product is +; but if the number of negative factors is odd, the sign of the product is -. That this is so, follows from the Rule given in § 23.

For example, the sign of 2x(-3)X(-4)X5 is t, since we here have two negative factors (or an even number); but the sign of 2x(-3)X(-4)X5X(-7) is –, since here we have three (or an odd number) of the negative factors.

NOTE 2. The sign X may be omitted in products containing literal numbers. Thus, instead of 2X (-a)X(-6)Xc we write 2(-a)(-6). This is merely an extension of what was said in § 3.

ORAL EXERCISES In each of the following, state first the sign of the result, then give the complete value. 1. 2X(-5)X(-8). 3. (-6) X(-2) X5. 2. 4X(-3) X2.

4. (-1) X(-2)X(-3) X(-4). 5. (-1)X(-1) X(-1) X(-1)X(-1). 6. (-2) X(-3) X4. 9. (-a)(-6)(–c);- abc. Ans. 7. {X(-{)X(-3). [Hint. See Note 2, in § 24.] 8. .1X(-.2)X(-.3). 10. (-2 a)(-6 b)(– 2 c). 11. (-a)b( -4 c).

14. xyz( abc)(-3 pqn). 12. (-x)(-y)(-2)(-w). 15. 3 p(-2 q)r2. 13. mn(–2 rs).

16. x(- y2)(-23).

WRITTEN EXERCISES If a=-1, b= -2, and c= -3, find the value of each of the following expressions. 1. 2 abc.

3. 2(–a)(-6). 2. 2(-a)bc.

4. 2(-a)(-6)(-c). [Hint. -a=-(-1)=1.) 5.2 ab+3 bc. 6. 2 ab- 3 bc. 12. V–2b. 16. a3b3c3. 7. ab+bc+ac. 13. V2b-3c. 17. a2b2c2 abc. 8. ab+bc - ac.

2 ab+c

18. ab2+boc. 9. ab-bc-ac. 3 c+b 10. a?+62+c2.

19. V4 a2b2+d2. 15.

abc+6 11. a3 +63+c3. 4c+66 20. C2 — 5 6. For further exercises on this topic, see Appendix, p. 292.

25. Powers and Exponents. We have already noted (see § 13) that x2 means x Xx and may be read “ x square.” It may also be read “x to the second power.” In the same way, x3 means x XxXx and may be read either “x cube" or“ x to the third power.”

We now note also that x4 means x X x Xx Xx and is read “x to the fourth power.” Again, xó means xXx Xx Xx Xx and is read “x to the fifth power,” etc. In all these cases the number denoting the power is called the exponent. Thus, 2 is the exponent in xạ, while 3 is the exponent in x3, etc.

26. Signs of the Powers of Numbers. In the first set below can be seen the powers of 2 from the first up to the sixth, while in the second set are the same powers of – 2:

21=2
22=2X2=4
23 = 2X2X2=8
24 = 2X2X2X2=16
25 = 2x2x2x2x2=32
26 = 2X2X2X2X2X2=64

(-2)= -2 (-2)2 =(-2) X(-2)= +4 (-2)3=(-2) X(-2)*(-2)= -8 (-2)4 = (-2) X(-2) X (-2) X(-2)= +16 (-2)5=(-2) X(-2)*(-2) X(-2)X(-2) = -32 (-2)6=(-2) X(-2) X(-2) X(-2) X(-2) X(-2) = +64

Observe that the results in the first set are all positive, while those in the second set are positive or negative according as the exponent used is an even or an odd number. This illustrates the following rule.

RULE OF SIGNS FOR POWERS. All powers of a positive number are positive. An even power of a negative number is positive, while an odd power of a negative number is negative.

ORAL EXERCISES In each of the following exercises, state first the sign of the result, then give the complete answer. 1. (-1) 8. (-6)3 15. (-a)3. 2. (-4)3. 9. (-1)6. 16. (-x)4. 3. (-1)5. 10. 33.

17. (-y)5. 4. 15.

11. (-3). 18. (-1)2X(-2)2. 5. (-6) 12. 34.

19. (-a)?(6). 6. (-1)? 13. (-3)4 20. xy( – 2)3. 7. (-5)3. 14. (-a)? 21. (-m)?( - n)3(-p)4.

27. Division of Positive and Negative Numbers. To say that 4X5=20 is the same as saying that 20:5=4.

Likewise, to say that 4X(-5) = –20 is the same as saying that (-20) --4= -5.

In the same way, from (-4) X 5 = – 20 we get (-20) + (-4) = 5, and finally from (-4) X(-5) = 20 we · have 20 =(-4)= -5.

These four cases illustrate all the possibilities in the division of one number (positive or negative) by another; from them we derive the following rule.

RULE OF DIVISION FOR Two NUMBERS. Divide the dividend by the divisor as in arithmetic, giving the positive sign to the quotient if both dividend and divisor have the same sign, but giving the negative sign to the quotient if the dividend and divisor have opposite signs.

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