Question

which expression is equivalent to \\(\frac{a^{2/3}}{b^{1/3}} + \frac{b^{2/3}}{a^{1/3}}\\)?

a. \\(\frac{a + b}{a^{1/3}b^{1/3}}\\)

b. \\(\frac{a^{2/3} + b^{2/3}}{a^{1/3} + b^{1/3}}\\)

c. \\(\frac{a + b}{a^{1/3} + b^{1/3}}\\)

d. \\(\frac{a^{2/9} + b^{2/9}}{a^{1/3}b^{1/3}}\\)

e. \\(\frac{a^{2/3}b^{1/3} + a^{1/3}b^{2/3}}{a^{1/3}b^{1/3}}\\)

Explanation:

Step1: Find a common denominator

The two fractions are $\frac{a^{2/3}}{b^{1/3}}$ and $\frac{b^{2/3}}{a^{1/3}}$. The common denominator of $b^{1/3}$ and $a^{1/3}$ is $a^{1/3}b^{1/3}$.

Step2: Rewrite each fraction with the common denominator

For the first fraction: $\frac{a^{2/3}}{b^{1/3}}=\frac{a^{2/3}\cdot a^{1/3}}{b^{1/3}\cdot a^{1/3}}=\frac{a^{(2/3 + 1/3)}}{a^{1/3}b^{1/3}}=\frac{a}{a^{1/3}b^{1/3}}$ (using the rule $x^m\cdot x^n=x^{m + n}$)
For the second fraction: $\frac{b^{2/3}}{a^{1/3}}=\frac{b^{2/3}\cdot b^{1/3}}{a^{1/3}\cdot b^{1/3}}=\frac{b^{(2/3+1/3)}}{a^{1/3}b^{1/3}}=\frac{b}{a^{1/3}b^{1/3}}$

Step3: Add the two fractions

$\frac{a}{a^{1/3}b^{1/3}}+\frac{b}{a^{1/3}b^{1/3}}=\frac{a + b}{a^{1/3}b^{1/3}}$ (since the denominators are the same, we add the numerators)

Answer:

A. $\frac{a + b}{a^{1/3}b^{1/3}}$