Certainly! Let's simplify the expression [tex]\(\sqrt[4]{\frac{16}{18}} \times \sqrt[5]{\frac{243}{32}}\)[/tex] step-by-step.
Step 1: Simplify the fractions inside the roots.
For the first term:
[tex]\[
\frac{16}{18} = \frac{8}{9}
\][/tex]
For the second term:
[tex]\[
\frac{243}{32}
\][/tex]
Step 2: Taking the roots of the simplified fractions.
1. For [tex]\(\sqrt[4]{\frac{8}{9}}\)[/tex]:
Recognize that we are taking the 4th root of the fraction.
[tex]\[
\sqrt[4]{\frac{8}{9}} \approx 0.9709835434146469
\][/tex]
2. For [tex]\(\sqrt[5]{\frac{243}{32}}\)[/tex]:
Recognize that we are taking the 5th root of the fraction.
[tex]\[
\sqrt[5]{\frac{243}{32}} = 1.5
\][/tex]
Step 3: Multiply the results of the roots.
[tex]\[
0.9709835434146469 \times 1.5 = 1.4564753151219703
\][/tex]
Thus, the simplified expression of [tex]\(\sqrt[4]{\frac{16}{18}} \times \sqrt[5]{\frac{243}{32}}\)[/tex] is approximately:
[tex]\[
1.4564753151219703
\][/tex]
