Solve frac{sqrt{8}+5}{sqrt{26}+13}

$$\frac{ \sqrt{ 8 } +5 }{ \sqrt{ 26 } +13 }$$

Evaluate

$$\frac{2\sqrt{2}}{11}-\frac{4\sqrt{13}}{143}-\frac{5\sqrt{26}}{143}+\frac{5}{11}\approx 0.432533216$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}+5}{\sqrt{26}+13}$$

Rationalize the denominator of \frac{2\sqrt{2}+5}{\sqrt{26}+13} by multiplying numerator and denominator by \sqrt{26}-13.

$$\frac{\left(2\sqrt{2}+5\right)\left(\sqrt{26}-13\right)}{\left(\sqrt{26}+13\right)\left(\sqrt{26}-13\right)}$$

Consider \left(\sqrt{26}+13\right)\left(\sqrt{26}-13\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{2}+5\right)\left(\sqrt{26}-13\right)}{\left(\sqrt{26}\right)^{2}-13^{2}}$$

Square \sqrt{26}. Square 13.

$$\frac{\left(2\sqrt{2}+5\right)\left(\sqrt{26}-13\right)}{26-169}$$

Subtract 169 from 26 to get -143.

$$\frac{\left(2\sqrt{2}+5\right)\left(\sqrt{26}-13\right)}{-143}$$

Apply the distributive property by multiplying each term of 2\sqrt{2}+5 by each term of \sqrt{26}-13.

$$\frac{2\sqrt{2}\sqrt{26}-26\sqrt{2}+5\sqrt{26}-65}{-143}$$

Factor 26=2\times 13. Rewrite the square root of the product \sqrt{2\times 13} as the product of square roots \sqrt{2}\sqrt{13}.

$$\frac{2\sqrt{2}\sqrt{2}\sqrt{13}-26\sqrt{2}+5\sqrt{26}-65}{-143}$$

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

$$\frac{2\times 2\sqrt{13}-26\sqrt{2}+5\sqrt{26}-65}{-143}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{13}-26\sqrt{2}+5\sqrt{26}-65}{-143}$$

Multiply both numerator and denominator by -1.

$$\frac{-4\sqrt{13}+26\sqrt{2}-5\sqrt{26}+65}{143}$$