Simplify ( fifth root of 2+ fifth root of 3)/(3 fifth root of 17)

$\frac{\sqrt[5]{2} + \sqrt[5]{3}}{3 \sqrt[5]{17}}$

Multiply $\frac{\sqrt[5]{2} + \sqrt[5]{3}}{3 \sqrt[5]{17}}$ by $\frac{{\sqrt[5]{17}}^{4}}{{\sqrt[5]{17}}^{4}}$ .

$\frac{\sqrt[5]{2} + \sqrt[5]{3}}{3 \sqrt[5]{17}} \cdot \frac{{\sqrt[5]{17}}^{4}}{{\sqrt[5]{17}}^{4}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[5]{2} + \sqrt[5]{3}}{3 \sqrt[5]{17}}$ by $\frac{{\sqrt[5]{17}}^{4}}{{\sqrt[5]{17}}^{4}}$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 \sqrt[5]{17} {\sqrt[5]{17}}^{4}}$

Move $\sqrt[5]{17}$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 (\sqrt[5]{17} {\sqrt[5]{17}}^{4})}$

Raise $\sqrt[5]{17}$ to the power of $1$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 ({\sqrt[5]{17}}^{1} {\sqrt[5]{17}}^{4})}$

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

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 {\sqrt[5]{17}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 {\sqrt[5]{17}}^{5}}$

Rewrite ${\sqrt[5]{17}}^{5}$ as $17$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{17}$ as $17^{\frac{1}{5}}$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 {(17^{\frac{1}{5}})}^{5}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 \cdot 17^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 \cdot 17^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 \cdot 17^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 \cdot 17^{1}}$

Evaluate the exponent.

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) {\sqrt[5]{17}}^{4}}{3 \cdot 17}$

Simplify the numerator.

Rewrite ${\sqrt[5]{17}}^{4}$ as ${(17^{4})}^{\frac{1}{5}}$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) \sqrt[5]{17^{4}}}{3 \cdot 17}$

Raise $17$ to the power of $4$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) \sqrt[5]{83521}}{3 \cdot 17}$

Multiply $3$ by $17$ .

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) \sqrt[5]{83521}}{51}$

The result can be shown in multiple forms.

Exact Form:

$\frac{(\sqrt[5]{2} + \sqrt[5]{3}) \sqrt[5]{83521}}{51}$

Decimal Form:

$0.45288817 \dots$