Solve frac{2-sqrt{5}}{7-sqrt{7}}*frac{7+sqrt{7}}{7+sqrt{7}}

$$\frac { 2 - \sqrt { 5 } } { 7 - \sqrt { 7 } } \times \frac { 7 + \sqrt { 7 } } { 7 + \sqrt { 7 } }$$

Evaluate

$$\frac{\sqrt{7}}{21}-\frac{\sqrt{5}}{6}-\frac{\sqrt{35}}{42}+\frac{1}{3}\approx -0.054215548$$

Solution Steps

Divide 7+\sqrt{7} by 7+\sqrt{7} to get 1.

$$\frac{2-\sqrt{5}}{7-\sqrt{7}}\times 1$$

Rationalize the denominator of \frac{2-\sqrt{5}}{7-\sqrt{7}} by multiplying numerator and denominator by 7+\sqrt{7}.

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

Consider \left(7-\sqrt{7}\right)\left(7+\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 7. Square \sqrt{7}.

$$\frac{\left(2-\sqrt{5}\right)\left(7+\sqrt{7}\right)}{49-7}\times 1$$

Subtract 7 from 49 to get 42.

$$\frac{\left(2-\sqrt{5}\right)\left(7+\sqrt{7}\right)}{42}\times 1$$

Express \frac{\left(2-\sqrt{5}\right)\left(7+\sqrt{7}\right)}{42}\times 1 as a single fraction.

$$\frac{\left(2-\sqrt{5}\right)\left(7+\sqrt{7}\right)}{42}$$

Apply the distributive property by multiplying each term of 2-\sqrt{5} by each term of 7+\sqrt{7}.

$$\frac{14+2\sqrt{7}-7\sqrt{5}-\sqrt{5}\sqrt{7}}{42}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{14+2\sqrt{7}-7\sqrt{5}-\sqrt{35}}{42}$$

Factor

$$\frac{2 \sqrt{7} + 14 - \sqrt{35} - 7 \sqrt{5}}{42} = -0.054215547701642004$$