Page:Elementary algebra (1896).djvu/22

9. The Sign of Continuation, $\cdots$, is read "and so on."

10. The Sign of Deduction, $\therefore$, is read "therefore" or "hence."

11. In arithmetical multiplication the order in which the factors of a product are written is immaterial. Thus

$3\times 4=4\times 3$

In like manner in Algebra $ab$ and $ba$ each denote the product of the two quantities represented by the letters $a$ and $b$, and have therefore the same value. Although it is immaterial in what order the factors of a product are written, it is usual to arrange them alphabetically. Fractional coefficients which are greater than unity are usually kept in the form of improper fractions.

Ex. If $a=6$, $x=7$, $z=5$, find the value of ${\tfrac {13}{10}}axz$.

Here ${\tfrac {13}{10}}axz={\frac {13}{10}}\times 6\times 7\times 5=273$.

EXAMPLES I. a.

If $a=7$, $b=2$, $c=1$, $x=5$, $y=3$, find the value of

1. $14x$.
2. $x^{3}$.

3. $3ax$.
4. $a^{3}$.

5. $5by$.
6. $b^{5}$.

7. $3b^{2}$.
8. $2ax$.

9. $6c^{4}$.
10. $4y^{3}$.

If $a=8$, $b=5$, $c=4$, $x=1$, $y=3$, find the value of

11. $3c^{2}$.
12. $7y^{3}$.
13. $5ab$.

14. $9xy$.
15. $8b^{3}$.
16. $3x^{5}$.

17. $x^{8}$.
18. $7y^{4}$.
19. $c^{x}$.

20. $b^{y}$.
21. $y^{c}$.
22. $x^{b}$.

23. $y^{b}$.
24. $a^{y}$.
25. $b^{x}$.

If $a=7$, $b=2$, $c=1$, $x=5$, $y=3$, find the value of

26. ${\tfrac {1}{4}}x^{4}$.
27. ${\tfrac {5}{12}}c^{3}$.

28. ${\tfrac {3}{8}}x^{3}$.
29. ${\tfrac {1}{10}}ax$.

30. $3^{x}$.
31. $2^{c}$.

32. $8^{b}$.
33. $7^{x}$.

34. ${\tfrac {7}{15}}acx$.
35. ${\tfrac {1}{8}}bcx$.

12. When several different quantities are multiplied together a notation similar to that of Art. 7 is adopted. Thus $aabbbbcddd$ is written $a^{2}b^{4}cd^{3}$. And conversely $7a^{3}cd^{2}$ has the same meaning as $7\times a\times a\times a\times c\times d\times d$.

Ex. 1. If $x=5$, $y=3$, find the value of $4x^{2}y^{3}$.

$4x^{2}y^{3}=4\times 5^{2}\times 3^{3}=4\times 25\times 27=2700$.