Solve frac{sqrt{59}*29-sqrt{5*29}}{sqrt{59*29}+sqrt{5*29}}

$$\frac { \sqrt { 59 } \cdot 29 - \sqrt { 5 \cdot 29 } } { \sqrt { 59 \cdot 29 } + \sqrt { 5 \cdot 29 } }$$

Evaluate

$$\frac{59\sqrt{29}+5-\sqrt{295}-\sqrt{8555}}{54}\approx 3.945479937$$

Solution Steps

Multiply 5 and 29 to get 145.

$$\frac{29\sqrt{59}-\sqrt{145}}{\sqrt{59\times 29}+\sqrt{5\times 29}}$$

Multiply 59 and 29 to get 1711.

$$\frac{29\sqrt{59}-\sqrt{145}}{\sqrt{1711}+\sqrt{5\times 29}}$$

Multiply 5 and 29 to get 145.

$$\frac{29\sqrt{59}-\sqrt{145}}{\sqrt{1711}+\sqrt{145}}$$

Rationalize the denominator of \frac{29\sqrt{59}-\sqrt{145}}{\sqrt{1711}+\sqrt{145}} by multiplying numerator and denominator by \sqrt{1711}-\sqrt{145}.

$$\frac{\left(29\sqrt{59}-\sqrt{145}\right)\left(\sqrt{1711}-\sqrt{145}\right)}{\left(\sqrt{1711}+\sqrt{145}\right)\left(\sqrt{1711}-\sqrt{145}\right)}$$

Consider \left(\sqrt{1711}+\sqrt{145}\right)\left(\sqrt{1711}-\sqrt{145}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(29\sqrt{59}-\sqrt{145}\right)\left(\sqrt{1711}-\sqrt{145}\right)}{\left(\sqrt{1711}\right)^{2}-\left(\sqrt{145}\right)^{2}}$$

Square \sqrt{1711}. Square \sqrt{145}.

$$\frac{\left(29\sqrt{59}-\sqrt{145}\right)\left(\sqrt{1711}-\sqrt{145}\right)}{1711-145}$$

Subtract 145 from 1711 to get 1566.

$$\frac{\left(29\sqrt{59}-\sqrt{145}\right)\left(\sqrt{1711}-\sqrt{145}\right)}{1566}$$

Apply the distributive property by multiplying each term of 29\sqrt{59}-\sqrt{145} by each term of \sqrt{1711}-\sqrt{145}.

$$\frac{29\sqrt{59}\sqrt{1711}-29\sqrt{59}\sqrt{145}-\sqrt{145}\sqrt{1711}+\left(\sqrt{145}\right)^{2}}{1566}$$

Factor 1711=59\times 29. Rewrite the square root of the product \sqrt{59\times 29} as the product of square roots \sqrt{59}\sqrt{29}.

$$\frac{29\sqrt{59}\sqrt{59}\sqrt{29}-29\sqrt{59}\sqrt{145}-\sqrt{145}\sqrt{1711}+\left(\sqrt{145}\right)^{2}}{1566}$$

Multiply \sqrt{59} and \sqrt{59} to get 59.

$$\frac{29\times 59\sqrt{29}-29\sqrt{59}\sqrt{145}-\sqrt{145}\sqrt{1711}+\left(\sqrt{145}\right)^{2}}{1566}$$

Multiply 29 and 59 to get 1711.

$$\frac{1711\sqrt{29}-29\sqrt{59}\sqrt{145}-\sqrt{145}\sqrt{1711}+\left(\sqrt{145}\right)^{2}}{1566}$$

To multiply \sqrt{59} and \sqrt{145}, multiply the numbers under the square root.

$$\frac{1711\sqrt{29}-29\sqrt{8555}-\sqrt{145}\sqrt{1711}+\left(\sqrt{145}\right)^{2}}{1566}$$

To multiply \sqrt{145} and \sqrt{1711}, multiply the numbers under the square root.

$$\frac{1711\sqrt{29}-29\sqrt{8555}-\sqrt{248095}+\left(\sqrt{145}\right)^{2}}{1566}$$

The square of \sqrt{145} is 145.

$$\frac{1711\sqrt{29}-29\sqrt{8555}-\sqrt{248095}+145}{1566}$$

Factor 248095=29^{2}\times 295. Rewrite the square root of the product \sqrt{29^{2}\times 295} as the product of square roots \sqrt{29^{2}}\sqrt{295}. Take the square root of 29^{2}.

$$\frac{1711\sqrt{29}-29\sqrt{8555}-29\sqrt{295}+145}{1566}$$