Simplify Expressions Using the Properties for Exponents.

$$\begin{gathered} \left(a\right)\frac{x^{18}}{x^3} \\ \left(b\right)\frac{5^{12}}{5^3} \\ \left(c\right)\frac{q^{12}}{q^{36}} \\ \left(d\right)\frac{{10}^2}{{10}^3} \end{gathered}$$

The given expression is $\frac{x^{18}}{x^3}$.

The main objective is to simplify the expression.

The quotient property of exponents are defined as follows:

$$\begin{gathered} \frac{a^m}{a^n}=a^{m-n},m>n \\ \frac{a^m}{a^n}=\frac{1}{a^{n-m}},n>m \end{gathered}$$.

Use the quotient property of exponents $\frac{a^m}{a^n}=a^{m-n},m>n$to simplify the expression.

localid="1645006171961" $\begin{array}{rcl}\frac{x^{18}}{x^3}& =& x^{18-3}\\ & =& x^{15}\end{array}$

So,$\begin{array}{rcl}\frac{x^{18}}{x^3}& =& x^{15}\end{array}$.

The given expression is $\frac{5^{12}}{5^3}$.

The main objective is to simplify the expression.

Use the quotient property $\frac{a^m}{a^n}=a^{m-n},m>n$, to simplify the expression.

$\begin{array}{rcl}\frac{5^{12}}{5^3}& =& 5^{12-3}\\ & =& 5^9\end{array}$

So,$\frac{5^{12}}{5^3}=5^9$.

The given expression is $\frac{q^{18}}{q^{36}}$.

The main objective is to simplify the expression.

Use the quotient property $\frac{a^m}{a^n}=\frac{1}{a^{n-m}},n>m$, to simplify the expression.

$\begin{array}{rcl}\frac{q^{18}}{q^{36}}& =& \frac{1}{q^{36-18}}\\ & =& \frac{1}{q^{18}}\end{array}$

So,$\begin{array}{rcl}\frac{q^{18}}{q^{36}}& =& \frac{1}{q^{18}}\end{array}$.

The given expression is$\frac{{10}^2}{{10}^3}=\frac{1}{10}$

The main objective is to simplify the expression.

Use the quotient property $\frac{a^m}{a^n}=\frac{1}{a^{n-m}},n>m$, to simplify the expression.

$\begin{array}{rcl}\frac{{10}^2}{{10}^3}& =& \frac{1}{{10}^{3-2}}\\ & =& \frac{1}{10}\end{array}$

So,$\frac{{10}^2}{{10}^3}=\frac{1}{10}$.