Solve frac{sqrt{5}+sqrt{2}}{7+2sqrt{70}}*frac{7-2sqrt{70}}{7-2sqrt{70}}

$$\frac { \sqrt { 5 } + \sqrt { 2 } } { 7 + 2 \sqrt { 70 } } \cdot \frac { 7 - 2 \sqrt { 70 } } { 7 - 2 \sqrt { 70 } }$$

Evaluate

$$\frac{4\sqrt{35}}{231}+\frac{10\sqrt{14}}{231}-\frac{\sqrt{2}}{33}-\frac{\sqrt{5}}{33}\approx 0.153804858$$

Solution Steps

Divide 7-2\sqrt{70} by 7-2\sqrt{70} to get 1.

$$\frac{\sqrt{5}+\sqrt{2}}{7+2\sqrt{70}}\times 1$$

Rationalize the denominator of \frac{\sqrt{5}+\sqrt{2}}{7+2\sqrt{70}} by multiplying numerator and denominator by 7-2\sqrt{70}.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{\left(7+2\sqrt{70}\right)\left(7-2\sqrt{70}\right)}\times 1$$

Consider \left(7+2\sqrt{70}\right)\left(7-2\sqrt{70}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{7^{2}-\left(2\sqrt{70}\right)^{2}}\times 1$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{49-\left(2\sqrt{70}\right)^{2}}\times 1$$

Expand \left(2\sqrt{70}\right)^{2}.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{49-2^{2}\left(\sqrt{70}\right)^{2}}\times 1$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{49-4\left(\sqrt{70}\right)^{2}}\times 1$$

The square of \sqrt{70} is 70.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{49-4\times 70}\times 1$$

Multiply 4 and 70 to get 280.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{49-280}\times 1$$

Subtract 280 from 49 to get -231.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{-231}\times 1$$

Express \frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{-231}\times 1 as a single fraction.

$$\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-2\sqrt{70}\right)}{-231}$$

Apply the distributive property by multiplying each term of \sqrt{5}+\sqrt{2} by each term of 7-2\sqrt{70}.

$$\frac{7\sqrt{5}-2\sqrt{5}\sqrt{70}+7\sqrt{2}-2\sqrt{2}\sqrt{70}}{-231}$$

Factor 70=5\times 14. Rewrite the square root of the product \sqrt{5\times 14} as the product of square roots \sqrt{5}\sqrt{14}.

$$\frac{7\sqrt{5}-2\sqrt{5}\sqrt{5}\sqrt{14}+7\sqrt{2}-2\sqrt{2}\sqrt{70}}{-231}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{7\sqrt{5}-2\times 5\sqrt{14}+7\sqrt{2}-2\sqrt{2}\sqrt{70}}{-231}$$

Multiply -2 and 5 to get -10.

$$\frac{7\sqrt{5}-10\sqrt{14}+7\sqrt{2}-2\sqrt{2}\sqrt{70}}{-231}$$

Factor 70=2\times 35. Rewrite the square root of the product \sqrt{2\times 35} as the product of square roots \sqrt{2}\sqrt{35}.

$$\frac{7\sqrt{5}-10\sqrt{14}+7\sqrt{2}-2\sqrt{2}\sqrt{2}\sqrt{35}}{-231}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{7\sqrt{5}-10\sqrt{14}+7\sqrt{2}-2\times 2\sqrt{35}}{-231}$$

Multiply -2 and 2 to get -4.

$$\frac{7\sqrt{5}-10\sqrt{14}+7\sqrt{2}-4\sqrt{35}}{-231}$$

Multiply both numerator and denominator by -1.

$$\frac{-7\sqrt{5}+10\sqrt{14}-7\sqrt{2}+4\sqrt{35}}{231}$$