Question

which expression is equivalent to $a^{1/3} + \frac{2}{a^{2/3}}$?\
a. $\frac{a^{2/9}+2}{a^{2/3}}$\
b. $\frac{2}{a^{1/3}}$\
c. $\frac{a^{1/3}+2}{a^{2/3}}$\
d. $\frac{a + 2}{a^{2/3}}$\
e. $\frac{a^{2/3}+2}{a}$

Explanation:

Step1: Rewrite term to common denominator

To combine the terms, rewrite $a^{1/3}$ with denominator $a^{2/3}$ using exponent rule $a^m \cdot a^n = a^{m+n}$:
$a^{1/3} = \frac{a^{1/3} \cdot a^{2/3}}{a^{2/3}} = \frac{a^{(1/3 + 2/3)}}{a^{2/3}} = \frac{a^1}{a^{2/3}} = \frac{a}{a^{2/3}}$

Step2: Combine the fractions

Add the two fractions with the common denominator:
$\frac{a}{a^{2/3}} + \frac{2}{a^{2/3}} = \frac{a + 2}{a^{2/3}}$

Answer:

D. $\frac{a+2}{a^{2/3}}$