Question

Add and subtractin radical 2 ^3√4+√6-√-5 ^3√4,+^3√6

Answer 1

To simplify the expression \( 2\sqrt[3]{4} + \sqrt{6} - \sqrt{-5}\sqrt[3]{4} + \sqrt[3]{6} \), we will organize the radicals and combine like terms, if possible.

1. **Calculate the cube roots**:
- \( \sqrt[3]{4} = 4^{1/3} \)
- \( \sqrt[3]{6} = 6^{1/3} \)

2. **Recognize that \( \sqrt{-5} \) means you are dealing with an imaginary number**:
- \( \sqrt{-5} = i\sqrt{5} \)

3. **Rewrite the expression**:
\[
2\sqrt[3]{4} + \sqrt{6} - i\sqrt{5}\sqrt[3]{4} + \sqrt[3]{6}
\]

4. **Group like terms**:
- The terms involving \( \sqrt[3]{4} \): \( 2\sqrt[3]{4} - i\sqrt{5}\sqrt[3]{4} \)
- The other terms: \( \sqrt{6} \) and \( \sqrt[3]{6} \)

5. **Factor out \( \sqrt[3]{4} \) from the first group**:
\[
\sqrt[3]{4}(2 - i\sqrt{5}) + \sqrt{6} + \sqrt[3]{6}
\]

Thus, the simplified expression is:
\[
\sqrt[3]{4}(2 - i\sqrt{5}) + \sqrt{6} + \sqrt[3]{6}
\]

This expression combines all terms appropriately, keeping in mind both real and imaginary parts.