Q. 5.25 - Intermediate Algebra

Question

Simplify each expression:

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

Step-by-Step Solution

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 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 x 15   x 10

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 x 15   x 10

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 6 14   6 5

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 6 14   6 5

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 x 18   x 22

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 x 18 &#160; x 22

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 &#160; 12 15 12 30

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 &#160; 12 15 12 30

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.

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