Solve 7/4-sqrt{8}+789/sqrt{485}

$$\frac{ 7 }{ 4- \sqrt{ 8 } } + \frac{ 789 }{ \sqrt{ 485 } }$$

Evaluate

$$\frac{7\sqrt{2}}{4}+\frac{789\sqrt{485}}{485}+\frac{7}{2}\approx 41.801518199$$

Short 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}.

$$\frac{7}{4-2\sqrt{2}}+\frac{789}{\sqrt{485}}$$

Rationalize the denominator of \frac{7}{4-2\sqrt{2}} by multiplying numerator and denominator by 4+2\sqrt{2}.

$$\frac{7\left(4+2\sqrt{2}\right)}{\left(4-2\sqrt{2}\right)\left(4+2\sqrt{2}\right)}+\frac{789}{\sqrt{485}}$$

Consider \left(4-2\sqrt{2}\right)\left(4+2\sqrt{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}.

$$\frac{7\left(4+2\sqrt{2}\right)}{4^{2}-\left(-2\sqrt{2}\right)^{2}}+\frac{789}{\sqrt{485}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{7\left(4+2\sqrt{2}\right)}{16-\left(-2\sqrt{2}\right)^{2}}+\frac{789}{\sqrt{485}}$$

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

$$\frac{7\left(4+2\sqrt{2}\right)}{16-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}+\frac{789}{\sqrt{485}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{7\left(4+2\sqrt{2}\right)}{16-4\left(\sqrt{2}\right)^{2}}+\frac{789}{\sqrt{485}}$$

The square of \sqrt{2} is 2.

$$\frac{7\left(4+2\sqrt{2}\right)}{16-4\times 2}+\frac{789}{\sqrt{485}}$$

Multiply 4 and 2 to get 8.

$$\frac{7\left(4+2\sqrt{2}\right)}{16-8}+\frac{789}{\sqrt{485}}$$

Subtract 8 from 16 to get 8.

$$\frac{7\left(4+2\sqrt{2}\right)}{8}+\frac{789}{\sqrt{485}}$$

Rationalize the denominator of \frac{789}{\sqrt{485}} by multiplying numerator and denominator by \sqrt{485}.

$$\frac{7\left(4+2\sqrt{2}\right)}{8}+\frac{789\sqrt{485}}{\left(\sqrt{485}\right)^{2}}$$

The square of \sqrt{485} is 485.

$$\frac{7\left(4+2\sqrt{2}\right)}{8}+\frac{789\sqrt{485}}{485}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 8 and 485 is 3880. Multiply \frac{7\left(4+2\sqrt{2}\right)}{8} times \frac{485}{485}. Multiply \frac{789\sqrt{485}}{485} times \frac{8}{8}.

$$\frac{485\times 7\left(4+2\sqrt{2}\right)}{3880}+\frac{8\times 789\sqrt{485}}{3880}$$

Since \frac{485\times 7\left(4+2\sqrt{2}\right)}{3880} and \frac{8\times 789\sqrt{485}}{3880} have the same denominator, add them by adding their numerators.

$$\frac{485\times 7\left(4+2\sqrt{2}\right)+8\times 789\sqrt{485}}{3880}$$

Do the multiplications in 485\times 7\left(4+2\sqrt{2}\right)+8\times 789\sqrt{485}.

$$\frac{13580+6790\sqrt{2}+6312\sqrt{485}}{3880}$$