Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume...

Step 1
For any rational exponent ${\displaystyle \frac{m}{n}}$ in lowest terms, where m and n are integers and ${\displaystyle n>0,}$ we define
${\displaystyle a^{\frac{m}{n}}={\left(\sqrt{n}\left\{a\right\}\right)}^m=\sqrt{n}\left\{a^m\right\}}$
If n is even, then we require that ${\displaystyle a\ge 0}$
Step 2
Consider the given expression,
${\displaystyle \frac{\sqrt{4}\left\{x^7\right\}}{\sqrt{4}\left\{x^3\right\}}}$
By using the law of exponents,
${\displaystyle \frac{\sqrt{4}\left\{x^7\right\}}{\sqrt{4}\left\{x^3\right\}}=\frac{x^{\frac{7}{4}}}{x^{\frac{3}{4}}}}$
Apply exponent rule: ${\displaystyle \frac{x^a}{x^b}=x^{a-b}}$, we get
${\displaystyle \frac{x^{\frac{7}{4}}}{x^{\frac{3}{4}}}=x^{\frac{7}{4}-\frac{3}{4}}}$
${\displaystyle =x^{\frac{4}{4}}}$
${\displaystyle =x}$
Final Statement:
The simplified form of ${\displaystyle \frac{\sqrt{4}\left\{x^7\right\}}{\sqrt{4}\left\{x^3\right\}}}$ is ${\displaystyle x}$.