Khaleesi Herbert

2020-12-30

Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$\frac{\sqrt{4}\left\{{x}^{7}\right\}}{\sqrt{4}\left\{{x}^{3}\right\}}$

Anonym

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

Do you have a similar question?