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ORAL EXERCISES

State the result of each of the following multiplications.

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1. a4 a5 NOTE.

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In the following exercises, we omit the dot. That is,

we write x3r instead of r3 · xo, etc.

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49. Product of Any Two Monomials. EXAMPLE. Multiply -3 a2x2 by 4 a3x4.

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Here we have first multiplied the -3 by 4 to obtain the new coefficient, -12, then we have multiplied a2 by a3, and x2 by x1, which, by Formula I, (§ 48) gives a3x. 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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Ans.

1. (3 a2bc2)X(-3 abc). Check when a=1, b=1, c=2. SOLUTION. Multiplying as in § 49, we obtain −9 a3b2c3. CHECK. When a=1, b=1, c=2, we see that 3 a2bc2=3×1×1×4 = 12; likewise that -3 abc=-3×1×1×2=-6, and -9 a3b2c3= -9X1X1X8=-72.

Since 12X(-6)=-72, our result checks.

2. (4 xyz2)×(-8 x2yz).

3. (2 a2bc2d) × (3 ab2c). d=1.

Check when x=1, y=2, z=2. Check, when a=1, b=1, c=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. (-xy3) · (xmyn). Check for x=2, y=2, a=2, b=2, m = 1, n = 1.

7. Simplify (-3 a2x) (5 a3x2y)+(2 m2n) (−3 n3q).

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

9. Simplify

(4 mngs) (−3 abqr) (2 brs) — (2 ghk) ( − 4 grs)(−3 abc).

50. Raising a Monomial to a Power. We shall note first the following examples :

(xy)2= (xy) · (xy) = xyxy=xxyy = x2y2.

(xy)3= (xy) · (xy) · (xy)=xxxyyy = x3y3.

(xy) = (xy) · (xy) · (xy) · (xy) = xxxxyyyy = x^y1.

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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

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(2 xy)2 = (2 x · y)2 = (2 x)2y2 = 22 x2y2 = 4 x2y2.

ORAL EXERCISES

State the result in each of the following exercises.

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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

S

A

B P

FIG. 19.

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 a2.]

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.

3

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:

6a+2b
3

18 a+6b Ans.

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