Solve A=2/sqrt{2}-3sqrt{8/9}-15sqrt{32/25}+10sqrt{2}

$$A = \frac { 2 } { \sqrt { 2 } } - 3 \sqrt { \frac { 8 } { 9 } } - 15 \sqrt { \frac { 32 } { 25 } } + 10 \sqrt { 2 }$$

Solve for A

$$A=-3\sqrt{2}\approx -4.242640687$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{2}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$A=\frac{2\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-3\sqrt{\frac{8}{9}}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

The square of \sqrt{2} is 2.

$$A=\frac{2\sqrt{2}}{2}-3\sqrt{\frac{8}{9}}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

Cancel out 2 and 2.

$$A=\sqrt{2}-3\sqrt{\frac{8}{9}}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

Rewrite the square root of the division \sqrt{\frac{8}{9}} as the division of square roots \frac{\sqrt{8}}{\sqrt{9}}.

$$A=\sqrt{2}-3\times \frac{\sqrt{8}}{\sqrt{9}}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

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}.

$$A=\sqrt{2}-3\times \frac{2\sqrt{2}}{\sqrt{9}}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

Calculate the square root of 9 and get 3.

$$A=\sqrt{2}-3\times \frac{2\sqrt{2}}{3}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

Express -3\times \frac{2\sqrt{2}}{3} as a single fraction.

$$A=\sqrt{2}+\frac{-3\times 2\sqrt{2}}{3}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

Cancel out 3 and 3.

$$A=\sqrt{2}-2\sqrt{2}-15\sqrt{\frac{32}{25}}+10\sqrt{2}$$

Rewrite the square root of the division \sqrt{\frac{32}{25}} as the division of square roots \frac{\sqrt{32}}{\sqrt{25}}.

$$A=\sqrt{2}-2\sqrt{2}-15\times \frac{\sqrt{32}}{\sqrt{25}}+10\sqrt{2}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$A=\sqrt{2}-2\sqrt{2}-15\times \frac{4\sqrt{2}}{\sqrt{25}}+10\sqrt{2}$$

Calculate the square root of 25 and get 5.

$$A=\sqrt{2}-2\sqrt{2}-15\times \frac{4\sqrt{2}}{5}+10\sqrt{2}$$

Cancel out 5, the greatest common factor in 15 and 5.

$$A=\sqrt{2}-2\sqrt{2}-3\times 4\sqrt{2}+10\sqrt{2}$$

Combine \sqrt{2} and 10\sqrt{2} to get 11\sqrt{2}.

$$A=11\sqrt{2}-2\sqrt{2}-3\times 4\sqrt{2}$$

Multiply -1 and 2 to get -2. Multiply -3 and 4 to get -12.

$$A=11\sqrt{2}-2\sqrt{2}-12\sqrt{2}$$

Combine 11\sqrt{2} and -2\sqrt{2} to get 9\sqrt{2}.

$$A=9\sqrt{2}-12\sqrt{2}$$

Combine 9\sqrt{2} and -12\sqrt{2} to get -3\sqrt{2}.

$$A=-3\sqrt{2}$$

Assign A

$$A≔-3\sqrt{2}$$