Final answer:
The expression \( \sqrt[4]{6} / \sqrt[3]{2} \) simplifies to \( 13.5^{1/12} \) by expressing the roots as fractional powers, finding a common denominator for the fractions, and then combining the powers using the quotient rule.
Step-by-step explanation:
The student is asking how to express \( \sqrt[4]{6} / \sqrt[3]{2} \) in an equivalent form. To simplify this expression, we must manipulate roots and powers. Using the property that \( x^{m/n} = \sqrt[n]{x^m} \), the fourth root of 6 can be written as \( 6^{1/4} \) and the cube root of 2 as \( 2^{1/3} \). Therefore, the given expression becomes:
\( \frac{\sqrt[4]{6}}{\sqrt[3]{2}} = \frac{6^{1/4}}{2^{1/3}} \)
To combine these expressions with different roots, we can find a common denominator for the fractional exponents. In this case, the common denominator of 4 and 3 is 12. We can then rewrite \( 6^{1/4} \) as \( 6^{3/12} \) and \( 2^{1/3} \) as \( 2^{4/12} \). Now, our expression is:
\( \frac{6^{3/12}}{2^{4/12}} \)
We can now use the quotient rule of exponents which states that \( a^{m/n} / b^{m/n} = (a/b)^{m/n} \) to simplify our expression:
\( \frac{6^{3/12}}{2^{4/12}} = (\frac{6^3}{2^4})^{1/12} \)
Calculating \( 6^3 \) and \( 2^4 \), we obtain 216 and 16, respectively. Thus, our final expression is
\( (\frac{216}{16})^{1/12} \)
Which can further simplify to:
\( 13.5^{1/12} \)
This final expression represents the same quantity as the original, but it is perhaps more suitable for further mathematical manipulation or understanding the relationship between the numbers.