Question

Evaluate: ${\left(\frac{16}{81}\right)}^{-\frac{3}{4}}\times {\left(\frac{49}{9}\right)}^{\frac{3}{2}}÷{\left(\frac{343}{216}\right)}^{\frac{2}{3}}$

Answer 1

To evaluate:

${\left(\frac{16}{81}\right)}^{-\frac{3}{4}}\times {\left(\frac{49}{9}\right)}^{\frac{3}{2}}÷{\left(\frac{343}{216}\right)}^{\frac{2}{3}}$

Now,

${\left(\frac{16}{81}\right)}^{-\frac{3}{4}}\times {\left(\frac{49}{9}\right)}^{\frac{3}{2}}÷{\left(\frac{343}{216}\right)}^{\frac{2}{3}}$

$={\left(\frac{81}{16}\right)}^{\frac{3}{4}}\times {\left(\frac{49}{9}\right)}^{\frac{3}{2}}÷{\left(\frac{343}{216}\right)}^{\frac{2}{3}}$

$={\left(\frac{3}{2}\right)}^{4\times \frac{3}{4}}\times {\left(\frac{7}{3}\right)}^{2\times \frac{3}{2}}÷{\left(\frac{7}{6}\right)}^{3\times \frac{2}{3}}$

$={\left(\frac{3}{2}\right)}^3\times {\left(\frac{7}{3}\right)}^3÷{\left(\frac{7}{6}\right)}^2$

$={\left(\frac{3}{2}\right)}^3\times {\left(\frac{7}{3}\right)}^3\times {\left(\frac{6}{7}\right)}^2$

$=\frac{3^3\times 7^3\times 6^2}{2^3\times 3^3\times 7^2}$

$=\frac{7\times 6^2}{2^3}$

$=\frac{7\times 2^2\times 3^2}{2^3}$

$=\frac{7\times 3^2}{2}$

$=\frac{63}{2}$

Hence, the required answer is $\frac{63}{2}$ .