Simplify (2a^(1/3)+a^(-1/3))/(a^(1/3))

$\frac{2 a^{\frac{1}{3}} + a^{- \frac{1}{3}}}{a^{\frac{1}{3}}}$

Step 1

Simplify the numerator.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 a^{\frac{1}{3}} + \frac{1}{a^{\frac{1}{3}}}}{a^{\frac{1}{3}}}$

To write $2 a^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{a^{\frac{1}{3}}}{a^{\frac{1}{3}}}$ .

$\frac{\frac{2 a^{\frac{1}{3}} a^{\frac{1}{3}}}{a^{\frac{1}{3}}} + \frac{1}{a^{\frac{1}{3}}}}{a^{\frac{1}{3}}}$

Combine the numerators over the common denominator.

$\frac{\frac{2 a^{\frac{1}{3}} a^{\frac{1}{3}} + 1}{a^{\frac{1}{3}}}}{a^{\frac{1}{3}}}$

Multiply $a^{\frac{1}{3}}$ by $a^{\frac{1}{3}}$ by adding the exponents.

Move $a^{\frac{1}{3}}$ .

$\frac{\frac{2 (a^{\frac{1}{3}} a^{\frac{1}{3}}) + 1}{a^{\frac{1}{3}}}}{a^{\frac{1}{3}}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 a^{\frac{1}{3} + \frac{1}{3}} + 1}{a^{\frac{1}{3}}}}{a^{\frac{1}{3}}}$

Combine the numerators over the common denominator.

$\frac{\frac{2 a^{\frac{1 + 1}{3}} + 1}{a^{\frac{1}{3}}}}{a^{\frac{1}{3}}}$

Add $1$ and $1$ .

$\frac{\frac{2 a^{\frac{2}{3}} + 1}{a^{\frac{1}{3}}}}{a^{\frac{1}{3}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{2 a^{\frac{2}{3}} + 1}{a^{\frac{1}{3}}} \cdot \frac{1}{a^{\frac{1}{3}}}$ .

Multiply $\frac{2 a^{\frac{2}{3}} + 1}{a^{\frac{1}{3}}}$ by $\frac{1}{a^{\frac{1}{3}}}$ .

$\frac{2 a^{\frac{2}{3}} + 1}{a^{\frac{1}{3}} a^{\frac{1}{3}}}$

Multiply $a^{\frac{1}{3}}$ by $a^{\frac{1}{3}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 a^{\frac{2}{3}} + 1}{a^{\frac{1}{3} + \frac{1}{3}}}$

Combine the numerators over the common denominator.

$\frac{2 a^{\frac{2}{3}} + 1}{a^{\frac{1 + 1}{3}}}$

Add $1$ and $1$ .

$\frac{2 a^{\frac{2}{3}} + 1}{a^{\frac{2}{3}}}$