Page:Elementary algebra (1896).djvu/23

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DEFINITIONS. SUBSTITUTIONS.

5

Ex. 2. If $a=4$ , $b=9$ , $x=6$ , find the value of ${\tfrac {8bx^{2}}{27a^{3}}}$ .

${\frac {8bx^{2}}{27a^{3}}}={\frac {8\times 9\times 6^{2}}{27\times 4^{3}}}={\frac {8\times 9\times 36}{27\times 64}}={\frac {3}{2}}=1{\frac {1}{2}}$ .

13. If one factor of a product is equal to 0, the product must be equal to 0, whatever values the other factors may have. A factor 0 is usually called a zero factor.

For instance, if $x=0$ , then $ab^{3}xy^{2}$ contains a zero factor. Therefore $ab^{3}xy^{2}=0$ when $x=0$ , whatever be the values of $a$ , $b$ , $y$ .

Again, if $c=0$ , then $c^{3}=0$ ; therefore $ab^{2}c^{2}$ , whatever values $a$ and $b$ may have.

Note. Every power 0 of is 0.

EXAMPLES I. a.

If $a=7$ , $b=2$ , $c=0$ , $x=5$ , $y=3$ , find the value of

1. $4ax^{2}$ .
2. $a^{3}b$ .

3. $8b^{2}y$ .
4. $3xy^{2}$ .

5. ${\tfrac {3}{4}}b^{2}x$ .
6. ${\tfrac {5}{6}}b^{3}y^{2}$ .

7. ${\tfrac {2}{5}}xy^{4}$ .
8. $a^{3}c$ .

9. $a^{2}cy$ .
10. $8x^{3}y$ .

If $a=2$ , $b=3$ , $c=1$ , $p=0$ , $q=4$ , $r=6$ , find the value of

11. ${\tfrac {3a^{2}r}{8b}}$ .
12. ${\tfrac {8ab^{2}}{9q^{2}}}$ .
13. ${\tfrac {6a^{3}c}{b^{2}}}$ .

14. ${\tfrac {4cr^{2}}{9a^{3}}}$ .
15. $3a^{2}b^{c}$ .
16. ${\tfrac {5}{6}}ba^{r}$ .

17. ${\tfrac {8b}{9a^{r}}}$ .
18. $5a^{b}c^{r}$ .
19. ${\tfrac {2a^{2}p}{7r}}$ .

20. $3^{a}2^{b}$ .
21. $2^{r}a^{5}$ .
22. $c^{b}b^{q}$ .

23. ${\tfrac {5a^{r}b^{q}}{64r^{a}}}$ .
24. ${\tfrac {27a^{q}}{32}}$ .
25. ${\tfrac {64}{q^{r}}}$ .

14. Definition. The square root of any proposed expression is that quantity whose square, or second power, is equal to the given expression. Thus the square root of $81$ is $9$ , because $9^{2}=81$ .

The square root of $a$ is denoted by ${\sqrt[{2}]{a}}$ , or more simply ${\sqrt {a}}$ .

Similarly the cube, fourth, fifth, etc., root of any expression is that quantity whose third, fourth, fifth, etc., power is equal to the given expression.

The roots are denoted by the symbols ${\sqrt[{3}]{}}$ , ${\sqrt[{4}]{}}$ , ${\sqrt[{5}]{}}$ , etc.

Examples:

${\sqrt[{3}]{27}}=3$ ; because $3^{3}=27$ .
${\sqrt[{5}]{32}}=2$ ; because $2^{5}=32$ .