To simplify the given expression [tex]\(\frac{\sqrt{2}}{\sqrt[3]{2}}\)[/tex], we'll express both the numerator and the denominator in terms of exponents with base 2.
1. Recall that [tex]\(\sqrt{2}\)[/tex] can be written as [tex]\(2^{\frac{1}{2}}\)[/tex]. This is because the square root of a number is equivalent to raising that number to the power of [tex]\( \frac{1}{2} \)[/tex].
So, [tex]\(\sqrt{2} = 2^{\frac{1}{2}}\)[/tex].
2. Similarly, [tex]\(\sqrt[3]{2}\)[/tex] can be written as [tex]\(2^{\frac{1}{3}}\)[/tex]. This is because the cube root of a number is equivalent to raising that number to the power of [tex]\( \frac{1}{3} \)[/tex].
So, [tex]\(\sqrt[3]{2} = 2^{\frac{1}{3}}\)[/tex].
3. Now, we rewrite the original fraction using these exponents:
[tex]\[\frac{\sqrt{2}}{\sqrt[3]{2}} = \frac{2^{\frac{1}{2}}}{2^{\frac{1}{3}}}\][/tex]
4. To simplify this expression, we use the property of exponents that allows us to subtract the exponents when dividing powers with the same base. Specifically, [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex].
Applying this property here:
[tex]\[\frac{2^{\frac{1}{2}}}{2^{\frac{1}{3}}} = 2^{\frac{1}{2} - \frac{1}{3}}\][/tex]
5. Next, we perform the subtraction of the exponents:
[tex]\[\frac{1}{2} - \frac{1}{3}\][/tex]
To subtract these fractions, we first find a common denominator. The least common multiple of 2 and 3 is 6.
Converting the fractions to have the same denominator:
[tex]\[\frac{1}{2} = \frac{3}{6}\][/tex]
[tex]\[\frac{1}{3} = \frac{2}{6}\][/tex]
Now, subtract:
[tex]\[\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\][/tex]
6. So, we have:
[tex]\[2^{\frac{1}{2} - \frac{1}{3}} = 2^{\frac{1}{6}}\][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{\sqrt{2}}{\sqrt[3]{2}}\)[/tex] is:
[tex]\[2^{\frac{1}{6}}\][/tex]
The correct answer is [tex]\(2^{\frac{1}{6}}\)[/tex].
So, the answer is:
[tex]\[2^{\frac{1}{6}}\][/tex]
