Exercise 62 Page 404

We are asked to find two monomials whose quotient is $24a^2{b}^3.$ We can break this down into $3$ simpler individual tasks.

1. Finding two numbers whose quotient is $24.$
2. Finding two powers with base $a$ whose quotient is $a^2.$
3. Finding two powers with base $b$ whose quotient is $b^3.$

For Task $1$ we can use any multiple of $24.$ For example, since $24\cdot 2=48,$ we know that $\frac{48}{2}=24.$ Then, for the other tasks, to find the required powers it is useful to recall the property for dividing same base powers.

To divide two powers with the same base, subtract the exponents. $$\begin{array}{c}\frac{a^m}{a^n}=a^{m-n}\end{array}$$

For Task $2$ we want to rewrite $a^2$ as the quotient of two same base powers $a^2=\frac{a^m}{a^n}.$ Then, according to this property we just need to chose two numbers $m$ and $n$ whose difference is $2.$ It is very similar for Task $3,$ with the only difference being that the final exponent for the power with base $b$ should be $3.$ We can find some examples in the table below.

$a^m$	$a^n$	$\frac{a^m}{a^n}$
$a^5$	$a^3$	$\frac{a^5}{a^3}=a^{5-3}=a^2$
$b^m$	$b^n$	$\frac{b^m}{b^n}$
$b^7$	$b^4$	$\frac{b^7}{b^4}=b^{7-4}=b^3$

Notice that these are just examples, as there are infinitely many possibilities satisfying these conditions. Now, if we put it all together, we obtain the quotient shown below. We have found the monomials $48a^5{b}^7$ and $2a^3{b}^4,$ whose quotient is $24a^2{b}^3,$ as required.