Question

Find the cube root of -343:3sqrt-343

Answer 1

To find the cube root of \(-343\), we can express it as:

\[
\sqrt[3]{-343}
\]

Since \(-343\) is a negative number, its cube root can also be negative. Moreover, we can recognize that:

\[
-343 = -7^3
\]

Thus, the cube root is:

\[
\sqrt[3]{-343} = \sqrt[3]{-7^3} = -7
\]

Now, you mentioned \(3\sqrt{-343}\). Let's find this quantity as well.

First, calculate \(\sqrt{-343}\):

\[
\sqrt{-343} = \sqrt{343} \cdot i
\]

The square root of \(343\) can be expressed as:

\[
\sqrt{343} = \sqrt{7^3} = 7\sqrt{7}
\]

Thus, we have:

\[
\sqrt{-343} = 7\sqrt{7} \cdot i
\]

Now, multiply by \(3\):

\[
3\sqrt{-343} = 3 \cdot 7\sqrt{7} \cdot i = 21\sqrt{7} \cdot i
\]

To summarize:
- The cube root of \(-343\) is \(-7\).
- The expression \(3\sqrt{-343}\) simplifies to \(21\sqrt{7} \cdot i\).