Solve A=7-frac{21/sqrt{3}-28+sqrt{14}}{sqrt{7}+sqrt{2}}

$$A = \frac { 7 - \frac { 21 } { \sqrt { 3 } } - 28 + \sqrt { 14 } } { \sqrt { 7 } + \sqrt { 2 } }$$

Solve for A

$$A=\frac{7\sqrt{6}+28\sqrt{2}-7\sqrt{21}-23\sqrt{7}}{5}\approx -7.237180415$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{21}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$A=\frac{7-\frac{21\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-28+\sqrt{14}}{\sqrt{7}+\sqrt{2}}$$

The square of \sqrt{3} is 3.

$$A=\frac{7-\frac{21\sqrt{3}}{3}-28+\sqrt{14}}{\sqrt{7}+\sqrt{2}}$$

Divide 21\sqrt{3} by 3 to get 7\sqrt{3}.

$$A=\frac{7-7\sqrt{3}-28+\sqrt{14}}{\sqrt{7}+\sqrt{2}}$$

Subtract 28 from 7 to get -21.

$$A=\frac{-21-7\sqrt{3}+\sqrt{14}}{\sqrt{7}+\sqrt{2}}$$

Rationalize the denominator of \frac{-21-7\sqrt{3}+\sqrt{14}}{\sqrt{7}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{7}-\sqrt{2}.

$$A=\frac{\left(-21-7\sqrt{3}+\sqrt{14}\right)\left(\sqrt{7}-\sqrt{2}\right)}{\left(\sqrt{7}+\sqrt{2}\right)\left(\sqrt{7}-\sqrt{2}\right)}$$

Consider \left(\sqrt{7}+\sqrt{2}\right)\left(\sqrt{7}-\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}.

$$A=\frac{\left(-21-7\sqrt{3}+\sqrt{14}\right)\left(\sqrt{7}-\sqrt{2}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{7}. Square \sqrt{2}.

$$A=\frac{\left(-21-7\sqrt{3}+\sqrt{14}\right)\left(\sqrt{7}-\sqrt{2}\right)}{7-2}$$

Subtract 2 from 7 to get 5.

$$A=\frac{\left(-21-7\sqrt{3}+\sqrt{14}\right)\left(\sqrt{7}-\sqrt{2}\right)}{5}$$

Apply the distributive property by multiplying each term of -21-7\sqrt{3}+\sqrt{14} by each term of \sqrt{7}-\sqrt{2}.

$$A=\frac{-21\sqrt{7}+21\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}-\left(-7\sqrt{3}\right)\sqrt{2}+\sqrt{14}\sqrt{7}-\sqrt{14}\sqrt{2}}{5}$$

Multiply -1 and -1 to get 1.

$$A=\frac{-21\sqrt{7}+21\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{3}\sqrt{2}+\sqrt{14}\sqrt{7}-\sqrt{14}\sqrt{2}}{5}$$

Factor 14=7\times 2. Rewrite the square root of the product \sqrt{7\times 2} as the product of square roots \sqrt{7}\sqrt{2}.

$$A=\frac{-21\sqrt{7}+21\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{3}\sqrt{2}+\sqrt{7}\sqrt{2}\sqrt{7}-\sqrt{14}\sqrt{2}}{5}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$A=\frac{-21\sqrt{7}+21\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{3}\sqrt{2}+7\sqrt{2}-\sqrt{14}\sqrt{2}}{5}$$

Combine 21\sqrt{2} and 7\sqrt{2} to get 28\sqrt{2}.

$$A=\frac{-21\sqrt{7}+28\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{3}\sqrt{2}-\sqrt{14}\sqrt{2}}{5}$$

Factor 14=2\times 7. Rewrite the square root of the product \sqrt{2\times 7} as the product of square roots \sqrt{2}\sqrt{7}.

$$A=\frac{-21\sqrt{7}+28\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{3}\sqrt{2}-\sqrt{2}\sqrt{7}\sqrt{2}}{5}$$

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

$$A=\frac{-21\sqrt{7}+28\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{3}\sqrt{2}-2\sqrt{7}}{5}$$

Combine -21\sqrt{7} and -2\sqrt{7} to get -23\sqrt{7}.

$$A=\frac{-23\sqrt{7}+28\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{3}\sqrt{2}}{5}$$

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

$$A=\frac{-23\sqrt{7}+28\sqrt{2}+\left(-7\sqrt{3}\right)\sqrt{7}+7\sqrt{6}}{5}$$

Multiply -1 and 7 to get -7.

$$A=\frac{-23\sqrt{7}+28\sqrt{2}-7\sqrt{3}\sqrt{7}+7\sqrt{6}}{5}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$A=\frac{-23\sqrt{7}+28\sqrt{2}-7\sqrt{21}+7\sqrt{6}}{5}$$

Divide each term of -23\sqrt{7}+28\sqrt{2}-7\sqrt{21}+7\sqrt{6} by 5 to get -\frac{23}{5}\sqrt{7}+\frac{28}{5}\sqrt{2}-\frac{7}{5}\sqrt{21}+\frac{7}{5}\sqrt{6}.

$$A=-\frac{23}{5}\sqrt{7}+\frac{28}{5}\sqrt{2}-\frac{7}{5}\sqrt{21}+\frac{7}{5}\sqrt{6}$$

Assign A

$$A≔\frac{7\sqrt{6}}{5}+\frac{84\sqrt{2}}{15}-\frac{7\sqrt{21}}{5}-\frac{69\sqrt{7}}{15}$$