Solve (3/2)^2div[1-3/2+sqrt{1/8+7/16}(1+frac{2}{3})^3]

$$( \frac { 3 } { 2 } ) ^ { 2 } \div [ 1 - \frac { 3 } { 2 } + \sqrt { \frac { 1 } { 8 } + \frac { 7 } { 16 } } ( 1 + \frac { 2 } { 3 } ) ^ { 3 } ]$$

Evaluate

$$\frac{81}{107}\approx 0.757009346$$

Solution Steps

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

$$\frac{\frac{9}{4}}{1-\frac{3}{2}+\sqrt{\frac{1}{8}+\frac{7}{16}}\left(1+\frac{2}{3}\right)^{3}}$$

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

$$\frac{\frac{9}{4}}{-\frac{1}{2}+\sqrt{\frac{1}{8}+\frac{7}{16}}\left(1+\frac{2}{3}\right)^{3}}$$

Add \frac{1}{8} and \frac{7}{16} to get \frac{9}{16}.

$$\frac{\frac{9}{4}}{-\frac{1}{2}+\sqrt{\frac{9}{16}}\left(1+\frac{2}{3}\right)^{3}}$$

Rewrite the square root of the division \frac{9}{16} as the division of square roots \frac{\sqrt{9}}{\sqrt{16}}. Take the square root of both numerator and denominator.

$$\frac{\frac{9}{4}}{-\frac{1}{2}+\frac{3}{4}\left(1+\frac{2}{3}\right)^{3}}$$

Add 1 and \frac{2}{3} to get \frac{5}{3}.

$$\frac{\frac{9}{4}}{-\frac{1}{2}+\frac{3}{4}\times \left(\frac{5}{3}\right)^{3}}$$

Calculate \frac{5}{3} to the power of 3 and get \frac{125}{27}.

$$\frac{\frac{9}{4}}{-\frac{1}{2}+\frac{3}{4}\times \frac{125}{27}}$$

Multiply \frac{3}{4} and \frac{125}{27} to get \frac{125}{36}.

$$\frac{\frac{9}{4}}{-\frac{1}{2}+\frac{125}{36}}$$

Add -\frac{1}{2} and \frac{125}{36} to get \frac{107}{36}.

$$\frac{\frac{9}{4}}{\frac{107}{36}}$$

Divide \frac{9}{4} by \frac{107}{36} by multiplying \frac{9}{4} by the reciprocal of \frac{107}{36}.

$$\frac{9}{4}\times \frac{36}{107}$$

Multiply \frac{9}{4} and \frac{36}{107} to get \frac{81}{107}.

$$\frac{81}{107}$$

Factor

$$\frac{3 ^ {4}}{107} = 0.7570093457943925$$