Chapter A.10, Problem 33AYU

To determine

To find: Simplify the expression $5\sqrt[3]{2}-2\sqrt[3]{54}$.

Answer to Problem 33AYU

The value of $5\sqrt[3]{2}-2\sqrt[3]{54}=-\sqrt[3]{2}$.

Explanation of Solution

Consider $5\sqrt[3]{2}-2\sqrt[3]{54}$,

$5\sqrt[3]{2}-2\sqrt[3]{54}=5{\left(2\right)}^{\frac{1}{3}}-2{\left(54\right)}^{\frac{1}{3}}=5{\left(2\right)}^{\frac{1}{3}}-2{\left(27\times 2\right)}^{\frac{1}{3}}$

$=5{\left(2\right)}^{\frac{1}{3}}-2{\left(3^3\times 2\right)}^{\frac{1}{3}}$

$=5{\left(2\right)}^{\frac{1}{3}}-2{\left(3^3\right)}^{\frac{1}{3}}{\left(2\right)}^{\frac{1}{3}}$

$={\left(2\right)}^{\frac{1}{3}}\left(5-2{\left(3^3\right)}^{\frac{1}{3}}\right)$

$={\left(2\right)}^{\frac{1}{3}}\left(-1\right)$

$=-\sqrt[3]{2}$