Solve sqrt{[(1/3+1)^{3}*(2-2/3)^4]^2:[(3/2-1/6)^5*(frac{7}{3}-1)^8]-frac{1}{2}}

$$\sqrt { [ ( \frac { 1 } { 3 } + 1 ) ^ { 3 } \cdot ( 2 - \frac { 2 } { 3 } ) ^ { 4 } ] ^ { 2 } : [ ( \frac { 3 } { 2 } - \frac { 1 } { 6 } ) ^ { 5 } \cdot ( \frac { 7 } { 3 } - 1 ) ^ { 8 } ] - \frac { 1 } { 2 } }$$

Evaluate

$$\frac{\sqrt{30}}{6}\approx 0.912870929$$

Solution Steps

Add \frac{1}{3} and 1 to get \frac{4}{3}.

$$\sqrt{\frac{\left(\left(\frac{4}{3}\right)^{3}\left(2-\frac{2}{3}\right)^{4}\right)^{2}}{\left(\frac{3}{2}-\frac{1}{6}\right)^{5}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Calculate \frac{4}{3} to the power of 3 and get \frac{64}{27}.

$$\sqrt{\frac{\left(\frac{64}{27}\left(2-\frac{2}{3}\right)^{4}\right)^{2}}{\left(\frac{3}{2}-\frac{1}{6}\right)^{5}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Subtract \frac{2}{3} from 2 to get \frac{4}{3}.

$$\sqrt{\frac{\left(\frac{64}{27}\times \left(\frac{4}{3}\right)^{4}\right)^{2}}{\left(\frac{3}{2}-\frac{1}{6}\right)^{5}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Calculate \frac{4}{3} to the power of 4 and get \frac{256}{81}.

$$\sqrt{\frac{\left(\frac{64}{27}\times \frac{256}{81}\right)^{2}}{\left(\frac{3}{2}-\frac{1}{6}\right)^{5}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Multiply \frac{64}{27} and \frac{256}{81} to get \frac{16384}{2187}.

$$\sqrt{\frac{\left(\frac{16384}{2187}\right)^{2}}{\left(\frac{3}{2}-\frac{1}{6}\right)^{5}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Calculate \frac{16384}{2187} to the power of 2 and get \frac{268435456}{4782969}.

$$\sqrt{\frac{\frac{268435456}{4782969}}{\left(\frac{3}{2}-\frac{1}{6}\right)^{5}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Subtract \frac{1}{6} from \frac{3}{2} to get \frac{4}{3}.

$$\sqrt{\frac{\frac{268435456}{4782969}}{\left(\frac{4}{3}\right)^{5}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Calculate \frac{4}{3} to the power of 5 and get \frac{1024}{243}.

$$\sqrt{\frac{\frac{268435456}{4782969}}{\frac{1024}{243}\left(\frac{7}{3}-1\right)^{8}}-\frac{1}{2}}$$

Subtract 1 from \frac{7}{3} to get \frac{4}{3}.

$$\sqrt{\frac{\frac{268435456}{4782969}}{\frac{1024}{243}\times \left(\frac{4}{3}\right)^{8}}-\frac{1}{2}}$$

Calculate \frac{4}{3} to the power of 8 and get \frac{65536}{6561}.

$$\sqrt{\frac{\frac{268435456}{4782969}}{\frac{1024}{243}\times \frac{65536}{6561}}-\frac{1}{2}}$$

Multiply \frac{1024}{243} and \frac{65536}{6561} to get \frac{67108864}{1594323}.

$$\sqrt{\frac{\frac{268435456}{4782969}}{\frac{67108864}{1594323}}-\frac{1}{2}}$$

Divide \frac{268435456}{4782969} by \frac{67108864}{1594323} by multiplying \frac{268435456}{4782969} by the reciprocal of \frac{67108864}{1594323}.

$$\sqrt{\frac{268435456}{4782969}\times \frac{1594323}{67108864}-\frac{1}{2}}$$

Multiply \frac{268435456}{4782969} and \frac{1594323}{67108864} to get \frac{4}{3}.

$$\sqrt{\frac{4}{3}-\frac{1}{2}}$$

Subtract \frac{1}{2} from \frac{4}{3} to get \frac{5}{6}.

$$\sqrt{\frac{5}{6}}$$

Rewrite the square root of the division \sqrt{\frac{5}{6}} as the division of square roots \frac{\sqrt{5}}{\sqrt{6}}.

$$\frac{\sqrt{5}}{\sqrt{6}}$$

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{5}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{5}\sqrt{6}}{6}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{30}}{6}$$