which expression is equivalent to $x^{2/3}+\frac{1}{x^{2/3}}$?\na. $\frac{x^{2/3}+1}{x^{2/3}}$\nb…

which expression is equivalent to $x^{2/3}+\frac{1}{x^{2/3}}$?\na. $\frac{x^{2/3}+1}{x^{2/3}}$\nb. $\frac{x^{4/9}+1}{x^{2/3}}$\nc. 1\nd. $\frac{x + 1}{x^{2/3}}$\ne. $\frac{x^{4/3}+1}{x^{2/3}}$

which expression is equivalent to $x^{2/3}+\frac{1}{x^{2/3}}$?\na. $\frac{x^{2/3}+1}{x^{2/3}}$\nb. $\frac{x^{4/9}+1}{x^{2/3}}$\nc. 1\nd. $\frac{x + 1}{x^{2/3}}$\ne. $\frac{x^{4/3}+1}{x^{2/3}}$

Answer

Explanation:

Step1: Find a common - denominator

The two terms are $x^{2/3}$ and $\frac{1}{x^{2/3}}$. The common denominator is $x^{2/3}$. Rewrite $x^{2/3}$ as $\frac{x^{2/3}\times x^{2/3}}{x^{2/3}}$. According to the rule of exponents $a^m\times a^n=a^{m + n}$, so $x^{2/3}\times x^{2/3}=x^{\frac{2}{3}+\frac{2}{3}}=x^{4/3}$. Then $x^{2/3}=\frac{x^{4/3}}{x^{2/3}}$.

Step2: Add the fractions

$x^{2/3}+\frac{1}{x^{2/3}}=\frac{x^{4/3}}{x^{2/3}}+\frac{1}{x^{2/3}}=\frac{x^{4/3}+1}{x^{2/3}}$.

Answer:

E. $\frac{x^{4/3}+1}{x^{2/3}}$