Solve 2.25-3sqrt{2*frac{13-2sqrt{13}}{13}-(frac{13-2sqrt{13}}{13})^2}+2*frac{13-2sqrt{13}}{13}

$$2.25 - 3 \sqrt { 2 \cdot \frac { 13 - 2 \sqrt { 13 } } { 13 } - ( \frac { 13 - 2 \sqrt { 13 } } { 13 } ) ^ { 2 } } + 2 \cdot \frac { 13 - 2 \sqrt { 13 } } { 13 }$$

Evaluate

$$4.25-\sqrt{13}\approx 0.644448725$$

Short Solution Steps

Express 2\times \frac{13-2\sqrt{13}}{13} as a single fraction.

$$2.25-3\sqrt{\frac{2\left(13-2\sqrt{13}\right)}{13}-\left(\frac{13-2\sqrt{13}}{13}\right)^{2}}+2\times \frac{13-2\sqrt{13}}{13}$$

To raise \frac{13-2\sqrt{13}}{13} to a power, raise both numerator and denominator to the power and then divide.

$$2.25-3\sqrt{\frac{2\left(13-2\sqrt{13}\right)}{13}-\frac{\left(13-2\sqrt{13}\right)^{2}}{13^{2}}}+2\times \frac{13-2\sqrt{13}}{13}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-2\sqrt{13}\right)^{2}.

$$2.25-3\sqrt{\frac{2\left(13-2\sqrt{13}\right)}{13}-\frac{169-52\sqrt{13}+4\left(\sqrt{13}\right)^{2}}{13^{2}}}+2\times \frac{13-2\sqrt{13}}{13}$$

The square of \sqrt{13} is 13.

$$2.25-3\sqrt{\frac{2\left(13-2\sqrt{13}\right)}{13}-\frac{169-52\sqrt{13}+4\times 13}{13^{2}}}+2\times \frac{13-2\sqrt{13}}{13}$$

Multiply 4 and 13 to get 52.

$$2.25-3\sqrt{\frac{2\left(13-2\sqrt{13}\right)}{13}-\frac{169-52\sqrt{13}+52}{13^{2}}}+2\times \frac{13-2\sqrt{13}}{13}$$

Add 169 and 52 to get 221.

$$2.25-3\sqrt{\frac{2\left(13-2\sqrt{13}\right)}{13}-\frac{221-52\sqrt{13}}{13^{2}}}+2\times \frac{13-2\sqrt{13}}{13}$$

Calculate 13 to the power of 2 and get 169.

$$2.25-3\sqrt{\frac{2\left(13-2\sqrt{13}\right)}{13}-\frac{221-52\sqrt{13}}{169}}+2\times \frac{13-2\sqrt{13}}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 13 and 169 is 169. Multiply \frac{2\left(13-2\sqrt{13}\right)}{13} times \frac{13}{13}.

$$2.25-3\sqrt{\frac{13\times 2\left(13-2\sqrt{13}\right)}{169}-\frac{221-52\sqrt{13}}{169}}+2\times \frac{13-2\sqrt{13}}{13}$$

Since \frac{13\times 2\left(13-2\sqrt{13}\right)}{169} and \frac{221-52\sqrt{13}}{169} have the same denominator, subtract them by subtracting their numerators.

$$2.25-3\sqrt{\frac{13\times 2\left(13-2\sqrt{13}\right)-\left(221-52\sqrt{13}\right)}{169}}+2\times \frac{13-2\sqrt{13}}{13}$$

Do the multiplications in 13\times 2\left(13-2\sqrt{13}\right)-\left(221-52\sqrt{13}\right).

$$2.25-3\sqrt{\frac{338-52\sqrt{13}-221+52\sqrt{13}}{169}}+2\times \frac{13-2\sqrt{13}}{13}$$

Do the calculations in 338-52\sqrt{13}-221+52\sqrt{13}.

$$2.25-3\sqrt{\frac{117}{169}}+2\times \frac{13-2\sqrt{13}}{13}$$

Express 2\times \frac{13-2\sqrt{13}}{13} as a single fraction.

$$2.25-3\sqrt{\frac{117}{169}}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

Reduce the fraction \frac{117}{169} to lowest terms by extracting and canceling out 13.

$$2.25-3\sqrt{\frac{9}{13}}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

Rewrite the square root of the division \sqrt{\frac{9}{13}} as the division of square roots \frac{\sqrt{9}}{\sqrt{13}}.

$$2.25-3\times \frac{\sqrt{9}}{\sqrt{13}}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

Calculate the square root of 9 and get 3.

$$2.25-3\times \frac{3}{\sqrt{13}}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

Rationalize the denominator of \frac{3}{\sqrt{13}} by multiplying numerator and denominator by \sqrt{13}.

$$2.25-3\times \frac{3\sqrt{13}}{\left(\sqrt{13}\right)^{2}}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

The square of \sqrt{13} is 13.

$$2.25-3\times \frac{3\sqrt{13}}{13}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

Express 3\times \frac{3\sqrt{13}}{13} as a single fraction.

$$2.25-\frac{3\times 3\sqrt{13}}{13}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

Multiply 3 and 3 to get 9.

$$2.25-\frac{9\sqrt{13}}{13}+\frac{2\left(13-2\sqrt{13}\right)}{13}$$

Use the distributive property to multiply 2 by 13-2\sqrt{13}.

$$2.25-\frac{9\sqrt{13}}{13}+\frac{26-4\sqrt{13}}{13}$$

Since -\frac{9\sqrt{13}}{13} and \frac{26-4\sqrt{13}}{13} have the same denominator, add them by adding their numerators.

$$2.25+\frac{-9\sqrt{13}+26-4\sqrt{13}}{13}$$

Do the calculations in -9\sqrt{13}+26-4\sqrt{13}.

$$2.25+\frac{-13\sqrt{13}+26}{13}$$

Divide each term of -13\sqrt{13}+26 by 13 to get -\sqrt{13}+2.

$$2.25-\sqrt{13}+2$$

Add 2.25 and 2 to get 4.25.

$$4.25-\sqrt{13}$$

Factor

$$\frac{17 - 4 \sqrt{13}}{4} = 0.6444487245360109$$