Sure, let's simplify the given expression step-by-step.
We start with the expression: [tex]\((\sqrt{2})(\sqrt[3]{2})\)[/tex].
1. Understanding the radicals:
- [tex]\(\sqrt{2}\)[/tex] can be written as [tex]\(2^{\frac{1}{2}}\)[/tex].
- [tex]\(\sqrt[3]{2}\)[/tex] can be written as [tex]\(2^{\frac{1}{3}}\)[/tex].
2. Rewriting the expression using exponents:
[tex]\[
(\sqrt{2})(\sqrt[3]{2}) = 2^{\frac{1}{2}} \times 2^{\frac{1}{3}}
\][/tex]
3. Using the property of exponents:
Recall that when multiplying exponents with the same base, we add the exponents:
[tex]\[
a^m \times a^n = a^{m+n}
\][/tex]
Applying this property to our expression:
[tex]\[
2^{\frac{1}{2}} \times 2^{\frac{1}{3}} = 2^{\frac{1}{2} + \frac{1}{3}}
\][/tex]
4. Adding the exponents:
We need a common denominator to add the fractions [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]:
- The common denominator for 2 and 3 is 6.
- Rewriting the exponents with the common denominator:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
[tex]\[
\frac{1}{3} = \frac{2}{6}
\][/tex]
Now we add the fractions:
[tex]\[
\frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\][/tex]
5. Simplified exponent:
[tex]\[
2^{\frac{1}{2} + \frac{1}{3}} = 2^{\frac{5}{6}}
\][/tex]
Thus, the expression [tex]\((\sqrt{2})(\sqrt[3]{2})\)[/tex] simplifies to [tex]\(2^{\frac{5}{6}}\)[/tex].
6. Verification with options:
- [tex]\(2^{\frac{1}{6}}\)[/tex]
- [tex]\(2^{\frac{2}{3}}\)[/tex]
- [tex]\(2^{\frac{5}{6}}\)[/tex]
- [tex]\(2^{\frac{7}{6}}\)[/tex]
From our simplification, the correct answer is [tex]\(\boxed{2^{\frac{5}{6}}}\)[/tex].
