Question

Perform the operations and simplify:
\root(3)()8 + 2 \root 50 - 5/root 18

Answer 1

To simplify the expression \( \sqrt[3]{8} + 2 \sqrt{50} - \frac{5}{\sqrt{18}} \), we will tackle each term separately.

1. **Calculate \(\sqrt[3]{8}\)**:
\[
\sqrt[3]{8} = 2
\]

2. **Calculate \(2 \sqrt{50}\)**:
First, simplify \(\sqrt{50}\):
\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\]
Now, calculate \(2\sqrt{50}\):
\[
2\sqrt{50} = 2 \cdot 5\sqrt{2} = 10\sqrt{2}
\]

3. **Calculate \(\frac{5}{\sqrt{18}}\)**:
First, simplify \(\sqrt{18}\):
\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
\]
Now, compute \(\frac{5}{\sqrt{18}}\):
\[
\frac{5}{\sqrt{18}} = \frac{5}{3\sqrt{2}}
\]
To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{2}\):
\[
\frac{5}{3\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{3 \cdot 2} = \frac{5\sqrt{2}}{6}
\]

Now we can combine our results:
\[
\sqrt[3]{8} + 2\sqrt{50} - \frac{5}{\sqrt{18}} = 2 + 10\sqrt{2} - \frac{5\sqrt{2}}{6}
\]

To combine \(10\sqrt{2}\) and \(-\frac{5\sqrt{2}}{6}\), we need a common denominator. The common denominator for 1 (which is \(6\)) is \(6\):
\[
10\sqrt{2} = \frac{60\sqrt{2}}{6}
\]
Now combine:
\[
\frac{60\sqrt{2}}{6} - \frac{5\sqrt{2}}{6} = \frac{60\sqrt{2} - 5\sqrt{2}}{6} = \frac{55\sqrt{2}}{6}
\]

So the final simplified result is:
\[
2 + \frac{55\sqrt{2}}{6}
\]

The final answer can be expressed as:
\[
\boxed{2 + \frac{55\sqrt{2}}{6}}
\]