Question

evaluate (8/9)^3/2×(8)^1/3​

Answer 1

Given:

${ (\frac{8}{9}) }^{ \frac{3}{2} } \times {(8)}^{ \frac{1}{3} }$

But,

$8 = {2}^{3} and \: 9 = {3}^{2}$

Putting the respective values, we get,

$= {(\frac{ {2}^{3} }{ {3}^{2} }) }^{ \frac{3}{2} } \times {( {2}^{3}) }^{ \frac{1}{3} }$

Now, we know that,

${( \frac{ {a}^{m} }{ {b}^{n} } )}^{x} = \frac{ {a}^{mx} }{ {b}^{nx} }$

So, according to this, we get,

$= \frac{ {2}^{ \frac{9}{2} } }{ {3}^{3} } \times {2}^{1}$

Now,we know that,

${a}^{m} \times {a}^{n } = {a}^{(m + n)}$

so, according to this, we get,

$= \frac{ {2}^{ (\frac{9}{2} + 1) } }{ {3}^{3} }$

$= \frac{ {2}^{ \frac{11}{2} } }{ {3}^{3} }$

$= \frac{ {2}^{5} \times {2}^{ \frac{1}{2} } }{ {3}^{3} }$

$= \frac{32 \sqrt{2} }{27}$