Question

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

Explanation:

Step1: Find common denominator

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

Step2: Rewrite fractions with common denominator

$\frac{a^{2/3} \cdot a^{1/3}}{a^{1/3}b^{1/3}} + \frac{b^{2/3} \cdot b^{1/3}}{a^{1/3}b^{1/3}}$

Step3: Simplify numerators using exponent rule

Use $x^m \cdot x^n = x^{m+n}$:
$\frac{a^{(2/3 + 1/3)}}{a^{1/3}b^{1/3}} + \frac{b^{(2/3 + 1/3)}}{a^{1/3}b^{1/3}} = \frac{a^{1}}{a^{1/3}b^{1/3}} + \frac{b^{1}}{a^{1/3}b^{1/3}}$

Step4: Combine the fractions

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

Answer:

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