Question

Simplify each expression:

$ⓐ\frac{x^{15}}{x^{10}}ⓑ\frac{6^{14}}{6^5}ⓒ\frac{x^{18}}{x^{22}}ⓓ\frac{{12}^{15}}{{12}^{30}}$

Short answer

The simplified expression of$\frac{x^{15}}{x^{10}}=x^5$

The simplified expression of $\frac{6^{14}}{6^5}=6^9$

The simplified expression of$\frac{x^{18}}{x^{22}}=\frac{1}{x^4}$

The simplified expression of$\frac{{12}^{15}}{{12}^{30}}=\frac{1}{{12}^{15}}$

Step-by-step solution

Step 1. Given Information 

In the given question we have to simplify each expression:

$ⓐ\frac{x^{15}}{x^{10}}ⓑ\frac{6^{14}}{6^5}ⓒ\frac{x^{18}}{x^{22}}ⓓ\frac{{12}^{15}}{{12}^{30}}$

Part(a) Step 1. Simplifying the expression x15 x10

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property,$\frac{a^m}{a^n}=a^{m-n}$when$m>n$

Part(a) Step 2. The given expression is x15 x10

Since $15>10$, there are more factors of$x$ in the numerator.
Use Quotient Property,

$\frac{x^{15}}{x^{10}}=x^{15-10}$

Simplify.

$\frac{x^{15}}{x^{10}}=x^5$

Part(b) Step 1. Simplifying the expression 614 65

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property,$\frac{a^m}{a^n}=a^{m-n}$when$m>n$

Part(b) Step 2. The given expression is 614 65

Since $14>5$, there are more factors of$x$ in the numerator.
Use Quotient Property,

$\frac{6^{14}}{6^5}=6^{14-5}$

Simplify.

$\frac{6^{14}}{6^5}=6^9$

Part(c) Step 1. Simplifying the expression x18 x22

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property, $\frac{a^m}{a^n}=\frac{1}{a^{m-n}}$when$m<n$

Part(c) Step 2. The given expression is x18 x22

Since $18<22$, there are more factors of$x$ in the denominator.
Use Quotient Property,

$\frac{x^{18}}{x^{22}}=\frac{1}{x^{22-18}}$

Simplify.

$\frac{x^{18}}{x^{22}}=\frac{1}{x^4}$

Part(d) Step 1. Simplifying the expression  12151230

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Using Quotient Property,$\frac{a^m}{a^n}=\frac{1}{a^{m-n}}$when$m<n$

Part(d) Step 2. The given expression is  12151230

Since $15<30$, there are more factors of$x$ in the denominator.
Use Quotient Property,

$\frac{{12}^{15}}{{12}^{30}}=\frac{1}{{12}^{30-15}}$

Simplify.

$\frac{{12}^{15}}{{12}^{30}}=\frac{1}{{12}^{15}}$