Solve frac{(3-2sqrt{2})^2-(3+2sqrt{2})^2}{(3-2sqrt{2})^2+(3+2sqrt{2})^2}

$$\frac { ( 3 - 2 \sqrt { 2 } ) ^ { 2 } - ( 3 + 2 \sqrt { 2 } ) ^ { 2 } } { ( 3 - 2 \sqrt { 2 } ) ^ { 2 } + ( 3 + 2 \sqrt { 2 } ) ^ { 2 } }$$

Evaluate

$$-\frac{12\sqrt{2}}{17}\approx -0.998268397$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-2\sqrt{2}\right)^{2}.

$$\frac{9-12\sqrt{2}+4\left(\sqrt{2}\right)^{2}-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{9-12\sqrt{2}+4\times 2-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{9-12\sqrt{2}+8-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Add 9 and 8 to get 17.

$$\frac{17-12\sqrt{2}-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+2\sqrt{2}\right)^{2}.

$$\frac{17-12\sqrt{2}-\left(9+12\sqrt{2}+4\left(\sqrt{2}\right)^{2}\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{17-12\sqrt{2}-\left(9+12\sqrt{2}+4\times 2\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{17-12\sqrt{2}-\left(9+12\sqrt{2}+8\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Add 9 and 8 to get 17.

$$\frac{17-12\sqrt{2}-\left(17+12\sqrt{2}\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

To find the opposite of 17+12\sqrt{2}, find the opposite of each term.

$$\frac{17-12\sqrt{2}-17-12\sqrt{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Subtract 17 from 17 to get 0.

$$\frac{-12\sqrt{2}-12\sqrt{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Combine -12\sqrt{2} and -12\sqrt{2} to get -24\sqrt{2}.

$$\frac{-24\sqrt{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-2\sqrt{2}\right)^{2}.

$$\frac{-24\sqrt{2}}{9-12\sqrt{2}+4\left(\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-24\sqrt{2}}{9-12\sqrt{2}+4\times 2+\left(3+2\sqrt{2}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{-24\sqrt{2}}{9-12\sqrt{2}+8+\left(3+2\sqrt{2}\right)^{2}}$$

Add 9 and 8 to get 17.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+2\sqrt{2}\right)^{2}.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+9+12\sqrt{2}+4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+9+12\sqrt{2}+4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+9+12\sqrt{2}+8}$$

Add 9 and 8 to get 17.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+17+12\sqrt{2}}$$

Add 17 and 17 to get 34.

$$\frac{-24\sqrt{2}}{34-12\sqrt{2}+12\sqrt{2}}$$

Combine -12\sqrt{2} and 12\sqrt{2} to get 0.

$$\frac{-24\sqrt{2}}{34}$$

Divide -24\sqrt{2} by 34 to get -\frac{12}{17}\sqrt{2}.

$$-\frac{12}{17}\sqrt{2}$$

Expand

$$-\frac{12 \sqrt{2}}{17} = -0.9982683969692436$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-2\sqrt{2}\right)^{2}.

$$\frac{9-12\sqrt{2}+4\left(\sqrt{2}\right)^{2}-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{9-12\sqrt{2}+4\times 2-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{9-12\sqrt{2}+8-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Add 9 and 8 to get 17.

$$\frac{17-12\sqrt{2}-\left(3+2\sqrt{2}\right)^{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+2\sqrt{2}\right)^{2}.

$$\frac{17-12\sqrt{2}-\left(9+12\sqrt{2}+4\left(\sqrt{2}\right)^{2}\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{17-12\sqrt{2}-\left(9+12\sqrt{2}+4\times 2\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{17-12\sqrt{2}-\left(9+12\sqrt{2}+8\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Add 9 and 8 to get 17.

$$\frac{17-12\sqrt{2}-\left(17+12\sqrt{2}\right)}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

To find the opposite of 17+12\sqrt{2}, find the opposite of each term.

$$\frac{17-12\sqrt{2}-17-12\sqrt{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Subtract 17 from 17 to get 0.

$$\frac{-12\sqrt{2}-12\sqrt{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Combine -12\sqrt{2} and -12\sqrt{2} to get -24\sqrt{2}.

$$\frac{-24\sqrt{2}}{\left(3-2\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-2\sqrt{2}\right)^{2}.

$$\frac{-24\sqrt{2}}{9-12\sqrt{2}+4\left(\sqrt{2}\right)^{2}+\left(3+2\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-24\sqrt{2}}{9-12\sqrt{2}+4\times 2+\left(3+2\sqrt{2}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{-24\sqrt{2}}{9-12\sqrt{2}+8+\left(3+2\sqrt{2}\right)^{2}}$$

Add 9 and 8 to get 17.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+\left(3+2\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+2\sqrt{2}\right)^{2}.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+9+12\sqrt{2}+4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+9+12\sqrt{2}+4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+9+12\sqrt{2}+8}$$

Add 9 and 8 to get 17.

$$\frac{-24\sqrt{2}}{17-12\sqrt{2}+17+12\sqrt{2}}$$

Add 17 and 17 to get 34.

$$\frac{-24\sqrt{2}}{34-12\sqrt{2}+12\sqrt{2}}$$

Combine -12\sqrt{2} and 12\sqrt{2} to get 0.

$$\frac{-24\sqrt{2}}{34}$$

Divide -24\sqrt{2} by 34 to get -\frac{12}{17}\sqrt{2}.

$$-\frac{12}{17}\sqrt{2}$$