Rational expression multiplied by a radical rational expression: (4 ÷ 9) × √(27 ÷ 8)

Question

Not sure where I need to go to complete:

$$\frac 49 \times \sqrt{\frac{27}8}.$$

I can get to $(2/3)^{5/2}$ but also not sure if this is correct.

Answer 1

$$\frac{4}{9} \cdot \sqrt{\frac{27}{8}}=\frac{4}{9}\cdot \frac{3}{2}\sqrt{\frac{3}{2}}=\frac{2}{3}\sqrt{\frac{3}{2}}=\frac{\sqrt{6}}{3}$$

But $\frac{2}{3}^{5/2}=\frac{4}{9}\sqrt{\frac{2}{3}}$ , which is obviously another number because $\sqrt{\frac{27}{8}}>1>\sqrt{\frac{2}{3}}$

Answer 2

You have

$$\frac 49\sqrt{\frac{27}8}=\frac 49\times\frac{3\sqrt{3}}{2\sqrt{2}}=\frac 23 \times \sqrt{\frac 32}=\sqrt{\frac 23}=\left(\frac 23\right)^{1/2}.$$