MULTIPLICATION AND DIVISION OF ALGEBRAIC NUMBERS

78. Index Law in Division.

The dividend contains all the factors of the divisor and of the quotient, and therefore the quotient contains the factors of the dividend that are not found in the divisor.

Thus, ${\displaystyle \frac{abc}{bc}}=a$, ${\displaystyle \frac{aabx}{ab}}=ax$, ${\displaystyle \frac{124abc}{-4ab}}=-31c$.

Divide $a^5$ by $a^2$, $a^6$ by $a^4$, $a^4$ by $a$. {4} \frac{a^{5}}{a^{2}} &= \frac{aaaaa}{aa} &&= aaa &&= a^{3} &&= a^{5-2}; \ \frac{a^{6}}{a^{4}} &= \frac{aaaaaa}{aaaa} &&= aa &&= a^{2} &&= a^{6-4}; \ \frac{a^{4}}{a} &= \frac{aaaa}{a} &&= aaa &&= a^{3} &&= a^{4-1}.

If $m$ and $n$ stand for any integers, and $m$ is greater than $n$, $a^m-a^n=a^{m-n}.$

The index of the quotient of two powers of the same letter is equal to the index of the letter in the dividend diminished by the index of the letter in the divisor.

79. Examples.

1. Divide $15xy$ by $5x$. $\frac{15xy}{5x}=\frac{3\times 5xy}{5x}=3y.$

Here we cancel the factors $5$ and $x$, which are common to the dividend and divisor.

2. Divide $-21a^2{b}^3$ by $3a{b}^2$. $\frac{-21a^2{b}^3}{3a{b}^2}=-7ab.$

3. Divide $54a^5{b}^{3}c$ by $-6a{b}^{2}c$. $\frac{54a^5{b}^{3}c}{-6a{b}^{2}c}=-9a^{4}b.$

4. Divide $-45x^4{y}^5{z}^7$ by $-15x^4{y}^5{z}^5$. $\frac{-45x^4{y}^5{z}^7}{-15x^4{y}^5{z}^5}=3z^2.$

5. Divide $-15a^3{b}^2{c}^3$ by $-60a^{2}b{c}^3$. $\frac{-15a^3{b}^2{c}^3}{-60a^{2}b{c}^3}=\frac{ab}{4}.$

Exercise 17.

Divide:

1. $x^3$ by $x$.

2. $21x^5$ by $7x^3$.

3. $35x^2$ by $-5x^2$.

4. $-42x^2$ by $6x^2$.

5. $-63x^5$ by $-9x$.

6. $-72x^3$ by $-8x^2$.

7. $-32a^2{b}^2$ by $8a{b}^2$.