Solve frac{sqrt{0.4*1.6}+sqrt{5.76:4}}{2/5}-5*(sqrt{(1+7/9)^2}-sqrt{(1-2/9)^2})

$$\frac { \sqrt { 0.4 \cdot 1.6 } + \sqrt { 5.76 : 4 } } { \frac { 2 } { 5 } } - 5 \cdot ( \sqrt { ( 1 + \frac { 7 } { 9 } ) ^ { 2 } } - \sqrt { ( 1 - \frac { 2 } { 9 } ) ^ { 2 } } )$$

Evaluate

$$0$$

Solution Steps

Multiply 0.4 and 1.6 to get 0.64.

$$\frac{\sqrt{0.64}+\sqrt{\frac{5.76}{4}}}{\frac{2}{5}}-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Calculate the square root of 0.64 and get 0.8.

$$\frac{0.8+\sqrt{\frac{5.76}{4}}}{\frac{2}{5}}-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Expand \frac{5.76}{4} by multiplying both numerator and the denominator by 100.

$$\frac{0.8+\sqrt{\frac{576}{400}}}{\frac{2}{5}}-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Reduce the fraction \frac{576}{400} to lowest terms by extracting and canceling out 16.

$$\frac{0.8+\sqrt{\frac{36}{25}}}{\frac{2}{5}}-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Rewrite the square root of the division \frac{36}{25} as the division of square roots \frac{\sqrt{36}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$\frac{0.8+\frac{6}{5}}{\frac{2}{5}}-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Add 0.8 and \frac{6}{5} to get 2.

$$\frac{2}{\frac{2}{5}}-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Divide 2 by \frac{2}{5} by multiplying 2 by the reciprocal of \frac{2}{5}.

$$2\times \frac{5}{2}-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Multiply 2 and \frac{5}{2} to get 5.

$$5-5\left(\sqrt{\left(1+\frac{7}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

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

$$5-5\left(\sqrt{\left(\frac{16}{9}\right)^{2}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Calculate \frac{16}{9} to the power of 2 and get \frac{256}{81}.

$$5-5\left(\sqrt{\frac{256}{81}}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Rewrite the square root of the division \frac{256}{81} as the division of square roots \frac{\sqrt{256}}{\sqrt{81}}. Take the square root of both numerator and denominator.

$$5-5\left(\frac{16}{9}-\sqrt{\left(1-\frac{2}{9}\right)^{2}}\right)$$

Subtract \frac{2}{9} from 1 to get \frac{7}{9}.

$$5-5\left(\frac{16}{9}-\sqrt{\left(\frac{7}{9}\right)^{2}}\right)$$

Calculate \frac{7}{9} to the power of 2 and get \frac{49}{81}.

$$5-5\left(\frac{16}{9}-\sqrt{\frac{49}{81}}\right)$$

Rewrite the square root of the division \frac{49}{81} as the division of square roots \frac{\sqrt{49}}{\sqrt{81}}. Take the square root of both numerator and denominator.

$$5-5\left(\frac{16}{9}-\frac{7}{9}\right)$$

Subtract \frac{7}{9} from \frac{16}{9} to get 1.

$$5-5\times 1$$

Multiply 5 and 1 to get 5.

$$5-5$$

Subtract 5 from 5 to get 0.

$$0$$

Factor

$$0$$