Solve frac{7sqrt{3}-5sqrt{2}}{sqrt{168}+sqrt{18}}=

$$\frac { 7 \sqrt { 3 } - 5 \sqrt { 2 } } { \sqrt { 168 } + \sqrt { 18 } } =$$

Evaluate

$$\frac{7\sqrt{14}}{25}-\frac{2\sqrt{21}}{15}-\frac{7\sqrt{6}}{50}+\frac{1}{5}\approx 0.293725412$$

Solution Steps

Factor 168=2^{2}\times 42. Rewrite the square root of the product \sqrt{2^{2}\times 42} as the product of square roots \sqrt{2^{2}}\sqrt{42}. Take the square root of 2^{2}.

$$\frac{7\sqrt{3}-5\sqrt{2}}{2\sqrt{42}+\sqrt{18}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{7\sqrt{3}-5\sqrt{2}}{2\sqrt{42}+3\sqrt{2}}$$

Rationalize the denominator of \frac{7\sqrt{3}-5\sqrt{2}}{2\sqrt{42}+3\sqrt{2}} by multiplying numerator and denominator by 2\sqrt{42}-3\sqrt{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{\left(2\sqrt{42}+3\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}$$

Consider \left(2\sqrt{42}+3\sqrt{2}\right)\left(2\sqrt{42}-3\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}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{\left(2\sqrt{42}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

Expand \left(2\sqrt{42}\right)^{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{2^{2}\left(\sqrt{42}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{4\left(\sqrt{42}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

The square of \sqrt{42} is 42.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{4\times 42-\left(3\sqrt{2}\right)^{2}}$$

Multiply 4 and 42 to get 168.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{168-\left(3\sqrt{2}\right)^{2}}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{168-3^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{168-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{168-9\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{168-18}$$

Subtract 18 from 168 to get 150.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(2\sqrt{42}-3\sqrt{2}\right)}{150}$$

Apply the distributive property by multiplying each term of 7\sqrt{3}-5\sqrt{2} by each term of 2\sqrt{42}-3\sqrt{2}.

$$\frac{14\sqrt{3}\sqrt{42}-21\sqrt{3}\sqrt{2}-10\sqrt{42}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{150}$$

Factor 42=3\times 14. Rewrite the square root of the product \sqrt{3\times 14} as the product of square roots \sqrt{3}\sqrt{14}.

$$\frac{14\sqrt{3}\sqrt{3}\sqrt{14}-21\sqrt{3}\sqrt{2}-10\sqrt{42}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{150}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{14\times 3\sqrt{14}-21\sqrt{3}\sqrt{2}-10\sqrt{42}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{150}$$

Multiply 14 and 3 to get 42.

$$\frac{42\sqrt{14}-21\sqrt{3}\sqrt{2}-10\sqrt{42}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{150}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{42\sqrt{14}-21\sqrt{6}-10\sqrt{42}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{150}$$

Factor 42=2\times 21. Rewrite the square root of the product \sqrt{2\times 21} as the product of square roots \sqrt{2}\sqrt{21}.

$$\frac{42\sqrt{14}-21\sqrt{6}-10\sqrt{2}\sqrt{21}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{150}$$

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

$$\frac{42\sqrt{14}-21\sqrt{6}-10\times 2\sqrt{21}+15\left(\sqrt{2}\right)^{2}}{150}$$

Multiply -10 and 2 to get -20.

$$\frac{42\sqrt{14}-21\sqrt{6}-20\sqrt{21}+15\left(\sqrt{2}\right)^{2}}{150}$$

The square of \sqrt{2} is 2.

$$\frac{42\sqrt{14}-21\sqrt{6}-20\sqrt{21}+15\times 2}{150}$$

Multiply 15 and 2 to get 30.

$$\frac{42\sqrt{14}-21\sqrt{6}-20\sqrt{21}+30}{150}$$