Solve sqrt[3]{27/8}-sqrt[2]{1-189/64}-sqrt{1-31/256}

$$\sqrt[ 3 ] { \frac { 27 } { 8 } } - \sqrt[ 2 ] { 1 - \frac { 189 } { 64 } } - \sqrt { 1 - \frac { 31 } { 256 } }$$

Evaluate (complex solution)

$$-\frac{5\sqrt{5}i}{8}+\frac{9}{16}\approx 0.5625-1.397542486i$$

Solution Steps

Calculate \sqrt[3]{\frac{27}{8}} and get \frac{3}{2}.

$$\frac{3}{2}-\sqrt[2]{1-\frac{189}{64}}-\sqrt{1-\frac{31}{256}}$$

Subtract \frac{189}{64} from 1 to get -\frac{125}{64}.

$$\frac{3}{2}-\sqrt[2]{-\frac{125}{64}}-\sqrt{1-\frac{31}{256}}$$

Subtract \frac{31}{256} from 1 to get \frac{225}{256}.

$$\frac{3}{2}-\sqrt[2]{-\frac{125}{64}}-\sqrt{\frac{225}{256}}$$

Rewrite the square root of the division \frac{225}{256} as the division of square roots \frac{\sqrt{225}}{\sqrt{256}}. Take the square root of both numerator and denominator.

$$\frac{3}{2}-\sqrt[2]{-\frac{125}{64}}-\frac{15}{16}$$

Subtract \frac{15}{16} from \frac{3}{2} to get \frac{9}{16}.

$$\frac{9}{16}-\sqrt[2]{-\frac{125}{64}}$$

Evaluate

$$\text{Indeterminate}$$

Solution Steps

Calculate \sqrt[3]{\frac{27}{8}} and get \frac{3}{2}.

$$\frac{3}{2}-\sqrt[2]{1-\frac{189}{64}}-\sqrt{1-\frac{31}{256}}$$

Subtract \frac{189}{64} from 1 to get -\frac{125}{64}.

$$\frac{3}{2}-\sqrt[2]{-\frac{125}{64}}-\sqrt{1-\frac{31}{256}}$$

Subtract \frac{31}{256} from 1 to get \frac{225}{256}.

$$\frac{3}{2}-\sqrt[2]{-\frac{125}{64}}-\sqrt{\frac{225}{256}}$$

Rewrite the square root of the division \frac{225}{256} as the division of square roots \frac{\sqrt{225}}{\sqrt{256}}. Take the square root of both numerator and denominator.

$$\frac{3}{2}-\sqrt[2]{-\frac{125}{64}}-\frac{15}{16}$$

Subtract \frac{15}{16} from \frac{3}{2} to get \frac{9}{16}.

$$\frac{9}{16}-\sqrt[2]{-\frac{125}{64}}$$

Factor (complex solution)

$$-\frac{5\sqrt{5}i}{8}+\frac{9}{16}$$