Solve frac{(sqrt{190}+sqrt{108})}{sqrt{10000}+sqrt{17}}

$$\frac { ( \sqrt { 190 } + \sqrt { 108 } ) } { \sqrt { 10000 } + \sqrt { 17 } }$$

Evaluate

$$\frac{100\sqrt{190}+600\sqrt{3}-\sqrt{3230}-6\sqrt{51}}{9983}\approx 0.232190093$$

Solution Steps

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

$$\frac{\sqrt{190}+6\sqrt{3}}{\sqrt{10000}+\sqrt{17}}$$

Calculate the square root of 10000 and get 100.

$$\frac{\sqrt{190}+6\sqrt{3}}{100+\sqrt{17}}$$

Rationalize the denominator of \frac{\sqrt{190}+6\sqrt{3}}{100+\sqrt{17}} by multiplying numerator and denominator by 100-\sqrt{17}.

$$\frac{\left(\sqrt{190}+6\sqrt{3}\right)\left(100-\sqrt{17}\right)}{\left(100+\sqrt{17}\right)\left(100-\sqrt{17}\right)}$$

Consider \left(100+\sqrt{17}\right)\left(100-\sqrt{17}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{190}+6\sqrt{3}\right)\left(100-\sqrt{17}\right)}{100^{2}-\left(\sqrt{17}\right)^{2}}$$

Square 100. Square \sqrt{17}.

$$\frac{\left(\sqrt{190}+6\sqrt{3}\right)\left(100-\sqrt{17}\right)}{10000-17}$$

Subtract 17 from 10000 to get 9983.

$$\frac{\left(\sqrt{190}+6\sqrt{3}\right)\left(100-\sqrt{17}\right)}{9983}$$

Apply the distributive property by multiplying each term of \sqrt{190}+6\sqrt{3} by each term of 100-\sqrt{17}.

$$\frac{100\sqrt{190}-\sqrt{190}\sqrt{17}+600\sqrt{3}-6\sqrt{3}\sqrt{17}}{9983}$$

To multiply \sqrt{190} and \sqrt{17}, multiply the numbers under the square root.

$$\frac{100\sqrt{190}-\sqrt{3230}+600\sqrt{3}-6\sqrt{3}\sqrt{17}}{9983}$$

To multiply \sqrt{3} and \sqrt{17}, multiply the numbers under the square root.

$$\frac{100\sqrt{190}-\sqrt{3230}+600\sqrt{3}-6\sqrt{51}}{9983}$$