Example 4
Completion requirements
| Example 4 |
Simplify the expression \(\left( {4\sqrt[3]{{15}} - 1} \right)\left( {\sqrt[3]{{25}} + 5} \right)\).
Step 1: Multiply the binomials using FOIL.
\(\begin{align}
\left( {4\sqrt[3]{{15}} - 1} \right)\left( {\sqrt[3]{{25}} + 5} \right) &= \left[ {4\sqrt[3]{{15}}\cdot \sqrt[3]{{25}}} \right] + \left[ {4\sqrt[3]{{15}}\cdot 5} \right] + \left[ {\left( { - 1} \right)\cdot \sqrt[3]{{25}}} \right] + \left[ {\left( { - 1} \right) \cdot 5} \right] \\
&= 4\sqrt[3]{{375}} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
\end{align}\)
Step 2: Check if any radicals can be further reduced.
\(\begin{align}
4\sqrt[3]{{375}} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 &= 4\sqrt[3]{{125\cdot 3}} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
&= 4\cdot 5\sqrt[3]{3} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
&= 20\sqrt[3]{3} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
\end{align}\)
\(\begin{align}
\left( {4\sqrt[3]{{15}} - 1} \right)\left( {\sqrt[3]{{25}} + 5} \right) &= \left[ {4\sqrt[3]{{15}}\cdot \sqrt[3]{{25}}} \right] + \left[ {4\sqrt[3]{{15}}\cdot 5} \right] + \left[ {\left( { - 1} \right)\cdot \sqrt[3]{{25}}} \right] + \left[ {\left( { - 1} \right) \cdot 5} \right] \\
&= 4\sqrt[3]{{375}} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
\end{align}\)
Step 2: Check if any radicals can be further reduced.
\(\begin{align}
4\sqrt[3]{{375}} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 &= 4\sqrt[3]{{125\cdot 3}} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
&= 4\cdot 5\sqrt[3]{3} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
&= 20\sqrt[3]{3} + 20\sqrt[3]{{15}} - \sqrt[3]{{25}} - 5 \\
\end{align}\)