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ORAL EXERCISES State the result of each of the following multiplications.

1. at · ab 2. b3 . 58 3. có c6 4. 23 • 26

Note. In the following exercises, we omit the dot. That is, we write x3x6 instead of x3 26, etc. 5. p12p3.

19. axa; a3x3. Ans. 6. død.

20. ayöa?y? 7. qq12.

21. aồyaRy4. 8. s783.

22. (-a)?a?. 9. 23218

(Hint. See g 26.] 10. ww6.

23. a^(-a). 11. a?am.

24. (-a)5(–a)4 12. aa.

25. (-ay2) · (ayt) 13. a"at.

26. (mon3)(-mona). 14. așa.

27. afto.q15. a*a".

28. x2x3x4. 16. q2tar.

[Hint. First write x2x3 = 25.] 17. Q4q27

29. zz22324. 18. q3tqšt

30. 22n 23nxon 49. Product of Any Two Monomials. EXAMPLE. Multiply – 3 a-x2 by 4 a3x4. SOLUTION.

-3 ar2

4 a’x4

- 12 a5x6. Ans. Here we have first multiplied the —3 by 4 to obtain the new coefficient, –12, then we have multiplied a2 by a', and x2 by x4, which, by Formula I, (§ 48) gives a5x6. So the answer is – 12 a5x6.

Similarly, in all cases we have the following rule.

RULE FOR MULTIPLYING MONOMIALS. To multiply one monomial by another, multiply the coefficients to obtain a new coefficient, then multiply the letters together, observing Formula I of § 48.

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WRITTEN EXERCISES Carry out the following indicated multiplications. 1. (3 a2bc2) X(-3 abc). Check when a=1, b=1, c=2. SOLUTION. Multiplying as in § 49, we obtain – 9 a3b2c3. Ans.

CHECK. When a=1, b=1, c=2, we see that 3 a2bc2=3X1X1X4 = 12; likewise that –3 abc=-3X1X1X2=-6, and -9 a3b2c3= - 9X1X1X8=–72.

Since 12X(-6)=–72, our result checks. 2. (4 xyz) X (-8 xyz). Check when x=1, y=2, 2=2.

3. (2 a2bc2d) X (3 ab?c). Check, when a=1, b=1, c=1, d=1.

4. (-4 a2bc2) X (2 ab) X(-3 ac). Check when a=1, b=2, c=1.

[Hint. Multiply the first two expressions together, then multiply what you get by the third expression.]

5. (2 abcd) · (4 ab) · (ac) · (bd). Check for a=2, b=1, c=1, d=1.

6. (— xoyb) · (zmy). Check for x=2, y=2, a=2, b=2, m=1, n=1.

7. Simplify (-3 a^2)(5 aco)+(2 mon)(-3 moq).

8. Check your answer for Ex. 7, by using a=1, x=1, y=2, m=2, n=1, q=2. 9. Simplify

(4 mnqs)(-3 abqr)(2 brs) (2 ghk)(4 qrs)(-3 abc). 50. Raising a Monomial to a Power. We shall note first the following examples :

(xy)2 = (xy) · (xy)=xyxy=xxyy=x_yo.
(wy)3 = (xy) · (xy) · (wy) = xxxyyy=x3ys.
(xy)4 = (xy) · (xy) · (xy) · (xy) = xxxxyyyy=x4y4.

(xg)'= ... =&g.
From these we infer the following general rule.

RULE FOR RAISING A PRODUCT TO A POWER. To raise the product of two numbers to any power, raise the two numbers separately to that power and take their product.

Stated as a formula, this becomes
Formula II. (xy)m=xmym.
Illustrations :

(2 x)2 =22 x2 =4 x2
(3 y)3=33 y:=27 y3.
(2 xy)2 = (2 x · y)2=(2 x)?y2 =22 xy2 =4 xạy.

ORAL EXERCISES State the result in each of the following exercises. 1. (2 x)3. 3. (5 s)3. 5. (2 mn)3. 7. (3 xyz)3. 2. (3 m)? 4. (3 ab)? 6. (8 abc)? 8. (2 abc)4. 9. (8 pqrs)?

11. (-2 mn)4. 13. (- 2 xạy2) 3. 10. (-3 ab)3.

12. (4 x2) 14. (7 a2bc)2. 16. (8 xyz)2+(3 mn)? 16. (-3 xy)3– (–2 xy)4+4(2 xy)?.

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Fig. 19.

17. In the figure are two squares, a side of the second one being twice as long as a side of the other. Show that the

area of the second is four times that of the first.

[Hint. Let a = the length of side of the first. Then 2 a = the length of side of the second. Now compare (2 a)2 with al.]

18. Show that if one cube has

its edges each twice as long as the

:: edges of a certain other cube, the volume of the one is eight times that of the other. See Ex. 17.

19. Compare the areas of two circles, one of which has a radius twice as great as the other. See Ex. 25, p. 22.

20. Compare the volumes of two spheres, one of which has a radius twice as great as the other. See Ex. 28, p. 23.

51. Multiplication of a Polynomial by a Monomial. In § 49 we saw how to multiply any monomial by another monomial. Let us now see how to multiply any polynomial by a monomial. If in arithmetic we wish to multiply 6 ft. and 2 in. by 3, we first multiply the 6 ft. by 3, getting 18 ft., then we multiply the 2 in. by 3, getting 6 in. The entire answer is 18 ft. and 6 in. This may be written as follows:

6 ft. +2 in.

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18 ft. +6 in. Ans. In the same way, if in algebra we wish to multiply the polynomial 6 a+2b by 3 we multiply each part separately and add the results, as indicated in the following scheme:

6 a+2b

3
18 a+6 b Ans.

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