Question

Combine and simplify the following radical expressions.

\[6\sqrt[3]{-16}+\frac{ 2 }{ 3 } \sqrt[3]{-54}-5\sqrt[3]{250}\]

Answer 1

I'm doing a review and its been a long time since I've done these. If anyone can guide me through the steps that would be great.

Answer 2

I would start by factoring the expressions under the radical sign

Answer 3

\(\large {
6\sqrt[3]{-16}+\frac{ 2 }{ 3 } \sqrt[3]{-54}-5\sqrt[3]{250}\qquad
\begin{cases}
16\to 2\cdot 2\cdot 2\cdot 2\to &2\cdot 2^3\\
54\to 2\cdot 3\cdot 3\cdot 3\to &2\cdot 3^3\\
250\to 2\cdot 5\cdot 5\cdot 5\to& 2\cdot 5^3
\end{cases}
\\ \quad \\
6\sqrt[3]{-16}+\cfrac{ 2 }{ 3 } \sqrt[3]{-54}-5\sqrt[3]{250}\implies ?
}\)

Answer 4

@perl Is this correct for after I factor it? \[12\sqrt[3]{2}+2\sqrt[3]{-2}-25\sqrt[3]{-2}\]

Answer 5

Whoops put negatives in wrong.
\[12\sqrt[3]{-2}+2\sqrt[3]{-2}-25\sqrt[3]{2}\]

Answer 6

\(\large {
6\sqrt[3]{-16}+\frac{ 2 }{ 3 } \sqrt[3]{-54}-5\sqrt[3]{250}\qquad
\begin{cases}
16\to 2\cdot 2\cdot 2\cdot 2\to &2\cdot 2^3\\
54\to 2\cdot 3\cdot 3\cdot 3\to &2\cdot 3^3\\
250\to 2\cdot 5\cdot 5\cdot 5\to& 2\cdot 5^3
\end{cases}
\\ \quad \\
6\sqrt[3]{-16}+\cfrac{ 2 }{ 3 } \sqrt[3]{-54}-5\sqrt[3]{250}
\\ \quad \\
6\sqrt[3]{-2^3\cdot 2}+\cfrac{ 2 }{ 3 }\sqrt[3]{-3^3\cdot 2}-5\sqrt[3]{-5^3\cdot 2}
\\ \quad \\

\begin{cases}
-2\cdot -2\cdot -2\to -2^3\\
-3\cdot -3\cdot -3\to -3^3\\
-5\cdot -5\cdot -5\to -5^3
\end{cases}
\\ \quad \\
6\sqrt[3]{-2^3\cdot 2}+\cfrac{ 2 }{ 3 }\sqrt[3]{-3^3\cdot 2}-5\sqrt[3]{-5^3\cdot 2}
\\ \quad \\
6\sqrt[3]{{\color{brown}{ -2}}^3} \cdot \sqrt[3]{2}+\cfrac{ 2 }{ 3 }\sqrt[3]{{\color{brown}{ -3}}^3}\sqrt[3]{2}-5\sqrt[3]{{\color{brown}{ -5}}^3} \cdot \sqrt[3]{ 2}
\\ \quad \\
6\cdot {\color{brown}{ -2}}\sqrt[3]{ 2}+\cfrac{ 2 }{ 3 }\cdot {\color{brown}{ -3}}\sqrt[3]{ 2}-5\cdot {\color{brown}{ -5}}\sqrt[3]{ 2}\implies ?
}\)