Q. 89 - Intermediate Algebra

Question

Simplify Expressions Using the Properties for Exponents.

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

Step-by-Step Solution

Answer

The simplified form is:

$$\begin{gathered} \left(a\right)\frac{x^{18}}{x^3}=x^{15} \\ \left(b\right)\frac{5^{12}}{5^3}=5^9 \\ \left(c\right)\frac{q^{18}}{q^{36}}=\frac{1}{q^{18}} \\ \left(d\right)\frac{{10}^2}{{10}^3}=\frac{1}{10} \end{gathered}$$

Part(a)-Step 1 Given information

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:

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

Part (a): Step 2: Simplify

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

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

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

Part (b): Step 1-Given information.

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

The main objective is to simplify the expression.

Part (b): Step 2: Simplify

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, $\frac{5^{12}}{5^3}=5^9$.

Part(c): Step 1- Given information

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

The main objective is to simplify the expression.

Part(c). Step 2. Simplify

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

Part(d). Step 1-Given information

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

The main objective is to simplify the expression.

Part(d). Step 2. simplify

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