Solve {left(3/2sqrt{16/6sqrt{2}}right)}^2

$${ \left( \frac{ 3 }{ 2 } \sqrt{ \frac{ 16 }{ 6 \sqrt{ 2 } } } \right) }^{ 2 }$$

Evaluate

$$3\sqrt{2}\approx 4.242640687$$

Solution Steps

Rationalize the denominator of \frac{16}{6\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\left(\frac{3}{2}\sqrt{\frac{16\sqrt{2}}{6\left(\sqrt{2}\right)^{2}}}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\left(\frac{3}{2}\sqrt{\frac{16\sqrt{2}}{6\times 2}}\right)^{2}$$

Cancel out 2\times 2 in both numerator and denominator.

$$\left(\frac{3}{2}\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}$$

Expand \left(\frac{3}{2}\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}.

$$\left(\frac{3}{2}\right)^{2}\left(\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{9}{4}\left(\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}$$

Calculate \sqrt{\frac{4\sqrt{2}}{3}} to the power of 2 and get \frac{4\sqrt{2}}{3}.

$$\frac{9}{4}\times \frac{4\sqrt{2}}{3}$$

Multiply \frac{9}{4} times \frac{4\sqrt{2}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9\times 4\sqrt{2}}{4\times 3}$$

Cancel out 3\times 4 in both numerator and denominator.

$$3\sqrt{2}$$

Expand

$$3 \sqrt{2} = 4.242640687$$

Solution Steps

Rationalize the denominator of \frac{16}{6\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\left(\frac{3}{2}\sqrt{\frac{16\sqrt{2}}{6\left(\sqrt{2}\right)^{2}}}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\left(\frac{3}{2}\sqrt{\frac{16\sqrt{2}}{6\times 2}}\right)^{2}$$

Cancel out 2\times 2 in both numerator and denominator.

$$\left(\frac{3}{2}\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}$$

Expand \left(\frac{3}{2}\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}.

$$\left(\frac{3}{2}\right)^{2}\left(\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{9}{4}\left(\sqrt{\frac{4\sqrt{2}}{3}}\right)^{2}$$

Calculate \sqrt{\frac{4\sqrt{2}}{3}} to the power of 2 and get \frac{4\sqrt{2}}{3}.

$$\frac{9}{4}\times \frac{4\sqrt{2}}{3}$$

Multiply \frac{9}{4} times \frac{4\sqrt{2}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9\times 4\sqrt{2}}{4\times 3}$$

Cancel out 3\times 4 in both numerator and denominator.

$$3\sqrt{2}$$