Answer
$$-7$$
Work Step by Step
RECALL:
(1) For $a\gt 0, b \gt0$, and an odd $n$, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.
(2) If $n$ is odd, $\sqrt[n]{a^n} = a$.
(3) If $n$ is even, $\sqrt[n]{a^n} = |a|$.
Using rule (1) above gives:
\begin{align*}
\sqrt[3]{-7} \cdot \sqrt[3]{49}&=\sqrt[3]{-7(49)}\\
&=\sqrt[3]{-343}\\
&=\sqrt[3]{-7(-7)(-7)}\\
&=\sqrt[3]{(-7)^3}\\
\end{align*}
Use rule (2) above to obtain:
\begin{align*}
\sqrt[3]{(-7)^3}&=-7
\end{align*}
