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

Skilled2020-12-31Added 108 answers

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 $x$ .

For any rational exponent

If n is even, then we require that

Step 2

Consider the given expression,

By using the law of exponents,

Apply exponent rule:

Final Statement:

The simplified form of