Solve =frac{6-2sqrt{5}}{sqrt{5}-2}*frac{sqrt{5}}{3-sqrt{5}}*10/5+sqrt{5}

$$= \frac { 6 - 2 \sqrt { 5 } } { \sqrt { 5 } - 2 } \cdot \frac { \sqrt { 5 } } { 3 - \sqrt { 5 } } \cdot \frac { 10 } { 5 + \sqrt { 5 } }$$

Evaluate

$$5\sqrt{5}+15\approx 26.180339887$$

Short Solution Steps

Rationalize the denominator of \frac{6-2\sqrt{5}}{\sqrt{5}-2} by multiplying numerator and denominator by \sqrt{5}+2.

$$\frac{\left(6-2\sqrt{5}\right)\left(\sqrt{5}+2\right)}{\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Consider \left(\sqrt{5}-2\right)\left(\sqrt{5}+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}.

$$\frac{\left(6-2\sqrt{5}\right)\left(\sqrt{5}+2\right)}{\left(\sqrt{5}\right)^{2}-2^{2}}\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Square \sqrt{5}. Square 2.

$$\frac{\left(6-2\sqrt{5}\right)\left(\sqrt{5}+2\right)}{5-4}\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Subtract 4 from 5 to get 1.

$$\frac{\left(6-2\sqrt{5}\right)\left(\sqrt{5}+2\right)}{1}\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Anything divided by one gives itself.

$$\left(6-2\sqrt{5}\right)\left(\sqrt{5}+2\right)\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Apply the distributive property by multiplying each term of 6-2\sqrt{5} by each term of \sqrt{5}+2.

$$\left(6\sqrt{5}+12-2\left(\sqrt{5}\right)^{2}-4\sqrt{5}\right)\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

The square of \sqrt{5} is 5.

$$\left(6\sqrt{5}+12-2\times 5-4\sqrt{5}\right)\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Multiply -2 and 5 to get -10.

$$\left(6\sqrt{5}+12-10-4\sqrt{5}\right)\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Subtract 10 from 12 to get 2.

$$\left(6\sqrt{5}+2-4\sqrt{5}\right)\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Combine 6\sqrt{5} and -4\sqrt{5} to get 2\sqrt{5}.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}}{3-\sqrt{5}}\times \frac{10}{5+\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{5}}{3-\sqrt{5}} by multiplying numerator and denominator by 3+\sqrt{5}.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}\times \frac{10}{5+\sqrt{5}}$$

Consider \left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}\times \frac{10}{5+\sqrt{5}}$$

Square 3. Square \sqrt{5}.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{9-5}\times \frac{10}{5+\sqrt{5}}$$

Subtract 5 from 9 to get 4.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{4}\times \frac{10}{5+\sqrt{5}}$$

Rationalize the denominator of \frac{10}{5+\sqrt{5}} by multiplying numerator and denominator by 5-\sqrt{5}.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{4}\times \frac{10\left(5-\sqrt{5}\right)}{\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)}$$

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

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{4}\times \frac{10\left(5-\sqrt{5}\right)}{5^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 5. Square \sqrt{5}.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{4}\times \frac{10\left(5-\sqrt{5}\right)}{25-5}$$

Subtract 5 from 25 to get 20.

$$\left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{4}\times \frac{10\left(5-\sqrt{5}\right)}{20}$$

Divide 10\left(5-\sqrt{5}\right) by 20 to get \frac{1}{2}\left(5-\sqrt{5}\right).

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

Use the distributive property to multiply \frac{1}{2} by 5-\sqrt{5}.

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

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Express \left(2\sqrt{5}+2\right)\times \frac{\sqrt{5}\left(3+\sqrt{5}\right)}{4} as a single fraction.

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

Use the distributive property to multiply \frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4} by \frac{5}{2}-\frac{1}{2}\sqrt{5}.

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

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

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

The square of \sqrt{5} is 5.

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

Multiply 2 and 5 to get 10.

$$\frac{\left(10+2\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4}\times \frac{5}{2}+\frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Apply the distributive property by multiplying each term of 10+2\sqrt{5} by each term of 3+\sqrt{5}.

$$\frac{30+10\sqrt{5}+6\sqrt{5}+2\left(\sqrt{5}\right)^{2}}{4}\times \frac{5}{2}+\frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Combine 10\sqrt{5} and 6\sqrt{5} to get 16\sqrt{5}.

$$\frac{30+16\sqrt{5}+2\left(\sqrt{5}\right)^{2}}{4}\times \frac{5}{2}+\frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

The square of \sqrt{5} is 5.

$$\frac{30+16\sqrt{5}+2\times 5}{4}\times \frac{5}{2}+\frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Multiply 2 and 5 to get 10.

$$\frac{30+16\sqrt{5}+10}{4}\times \frac{5}{2}+\frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Add 30 and 10 to get 40.

$$\frac{40+16\sqrt{5}}{4}\times \frac{5}{2}+\frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Multiply \frac{40+16\sqrt{5}}{4} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{\left(2\sqrt{5}+2\right)\sqrt{5}\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

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

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{\left(2\left(\sqrt{5}\right)^{2}+2\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

The square of \sqrt{5} is 5.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{\left(2\times 5+2\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Multiply 2 and 5 to get 10.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{\left(10+2\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Apply the distributive property by multiplying each term of 10+2\sqrt{5} by each term of 3+\sqrt{5}.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{30+10\sqrt{5}+6\sqrt{5}+2\left(\sqrt{5}\right)^{2}}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Combine 10\sqrt{5} and 6\sqrt{5} to get 16\sqrt{5}.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{30+16\sqrt{5}+2\left(\sqrt{5}\right)^{2}}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

The square of \sqrt{5} is 5.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{30+16\sqrt{5}+2\times 5}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Multiply 2 and 5 to get 10.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{30+16\sqrt{5}+10}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Add 30 and 10 to get 40.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{40+16\sqrt{5}}{4}\left(-\frac{1}{2}\right)\sqrt{5}$$

Multiply \frac{40+16\sqrt{5}}{4} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{-\left(40+16\sqrt{5}\right)}{4\times 2}\sqrt{5}$$

Express \frac{-\left(40+16\sqrt{5}\right)}{4\times 2}\sqrt{5} as a single fraction.

$$\frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2}+\frac{-\left(40+16\sqrt{5}\right)\sqrt{5}}{4\times 2}$$

Since \frac{\left(40+16\sqrt{5}\right)\times 5}{4\times 2} and \frac{-\left(40+16\sqrt{5}\right)\sqrt{5}}{4\times 2} have the same denominator, add them by adding their numerators.

$$\frac{\left(40+16\sqrt{5}\right)\times 5-\left(40+16\sqrt{5}\right)\sqrt{5}}{4\times 2}$$

Do the multiplications in \left(40+16\sqrt{5}\right)\times 5-\left(40+16\sqrt{5}\right)\sqrt{5}.

$$\frac{200+80\sqrt{5}-40\sqrt{5}-80}{4\times 2}$$

Do the calculations in 200+80\sqrt{5}-40\sqrt{5}-80.

$$\frac{120+40\sqrt{5}}{4\times 2}$$

Expand 4\times 2.

$$\frac{120+40\sqrt{5}}{8}$$

Factor

$$5 {(\sqrt{5} + 3)} = 26.180339887$$