To simplify the expression [tex]\((\sqrt{2})(\sqrt[3]{2})\)[/tex], we need to express both terms with the same base and deal with the exponents.
First, recall the meanings of the terms:
- [tex]\(\sqrt{2}\)[/tex] is the same as [tex]\(2^{1/2}\)[/tex].
- [tex]\(\sqrt[3]{2}\)[/tex] is the same as [tex]\(2^{1/3}\)[/tex].
The given product is:
[tex]\[
(\sqrt{2})(\sqrt[3]{2}) = (2^{1/2})(2^{1/3})
\][/tex]
When multiplying exponential expressions with the same base, we add their exponents:
[tex]\[
2^{1/2} \times 2^{1/3} = 2^{1/2 + 1/3}
\][/tex]
Next, we need to add the fractions [tex]\(1/2\)[/tex] and [tex]\(1/3\)[/tex]. To do this, we find a common denominator. The least common multiple of 2 and 3 is 6. Converting each fraction to have this common denominator:
[tex]\[
1/2 = 3/6 \quad \text{and} \quad 1/3 = 2/6
\][/tex]
Now, adding these fractions together:
[tex]\[
\frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\][/tex]
So the exponent sum is [tex]\(5/6\)[/tex], and the simplified product is:
[tex]\[
2^{1/2 + 1/3} = 2^{5/6}
\][/tex]
Therefore, the simplified form of [tex]\((\sqrt{2})(\sqrt[3]{2})\)[/tex] is:
[tex]\[
\boxed{2^{\frac{5}{6}}}
\][/tex]
