Solve frac{5(3+5)*frac{sqrt{2}}{2}}{frac{sqrt{5}+sqrt{2}}{4}}

$$\frac { 5 ( 3 + 5 ) \times \frac { \sqrt { 2 } } { 2 } } { \frac { \sqrt { 5 } + \sqrt { 2 } } { 4 } }$$

Evaluate

$$\frac{80\left(\sqrt{10}-2\right)}{3}\approx 30.994070938$$

Solution Steps

Add 3 and 5 to get 8.

$$\frac{5\times 8\times \frac{\sqrt{2}}{2}}{\frac{\sqrt{5}+\sqrt{2}}{4}}$$

Multiply 5 and 8 to get 40.

$$\frac{40\times \frac{\sqrt{2}}{2}}{\frac{\sqrt{5}+\sqrt{2}}{4}}$$

Cancel out 2, the greatest common factor in 40 and 2.

$$\frac{20\sqrt{2}}{\frac{\sqrt{5}+\sqrt{2}}{4}}$$

Divide 20\sqrt{2} by \frac{\sqrt{5}+\sqrt{2}}{4} by multiplying 20\sqrt{2} by the reciprocal of \frac{\sqrt{5}+\sqrt{2}}{4}.

$$\frac{20\sqrt{2}\times 4}{\sqrt{5}+\sqrt{2}}$$

Rationalize the denominator of \frac{20\sqrt{2}\times 4}{\sqrt{5}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{2}.

$$\frac{20\sqrt{2}\times 4\left(\sqrt{5}-\sqrt{2}\right)}{\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)}$$

Consider \left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{20\sqrt{2}\times 4\left(\sqrt{5}-\sqrt{2}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{5}. Square \sqrt{2}.

$$\frac{20\sqrt{2}\times 4\left(\sqrt{5}-\sqrt{2}\right)}{5-2}$$

Subtract 2 from 5 to get 3.

$$\frac{20\sqrt{2}\times 4\left(\sqrt{5}-\sqrt{2}\right)}{3}$$

Multiply 20 and 4 to get 80.

$$\frac{80\sqrt{2}\left(\sqrt{5}-\sqrt{2}\right)}{3}$$

Use the distributive property to multiply 80\sqrt{2} by \sqrt{5}-\sqrt{2}.

$$\frac{80\sqrt{2}\sqrt{5}-80\left(\sqrt{2}\right)^{2}}{3}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{80\sqrt{10}-80\left(\sqrt{2}\right)^{2}}{3}$$

The square of \sqrt{2} is 2.

$$\frac{80\sqrt{10}-80\times 2}{3}$$

Multiply -80 and 2 to get -160.

$$\frac{80\sqrt{10}-160}{3}$$