Solve frac{4(1+sqrt{2})(1-sqrt{2})}{3+2sqrt{2}}

$$\frac { 4 ( 1 + \sqrt { 2 } ) ( 1 - \sqrt { 2 } ) } { 3 + 2 \sqrt { 2 } }$$

Evaluate

$$4\left(2\sqrt{2}-3\right)\approx -0.686291501$$

Solution Steps

Rationalize the denominator of \frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}{3+2\sqrt{2}} by multiplying numerator and denominator by 3-2\sqrt{2}.

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}$$

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

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{3^{2}-\left(2\sqrt{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{9-\left(2\sqrt{2}\right)^{2}}$$

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

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{9-2^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{9-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{9-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{9-8}$$

Subtract 8 from 9 to get 1.

$$\frac{4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)}{1}$$

Anything divided by one gives itself.

$$4\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)$$

Use the distributive property to multiply 4 by 1+\sqrt{2}.

$$\left(4+4\sqrt{2}\right)\left(1-\sqrt{2}\right)\left(3-2\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of 4+4\sqrt{2} by each term of 1-\sqrt{2}.

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

Combine -4\sqrt{2} and 4\sqrt{2} to get 0.

$$\left(4-4\left(\sqrt{2}\right)^{2}\right)\left(3-2\sqrt{2}\right)$$

The square of \sqrt{2} is 2.

$$\left(4-4\times 2\right)\left(3-2\sqrt{2}\right)$$

Multiply -4 and 2 to get -8.

$$\left(4-8\right)\left(3-2\sqrt{2}\right)$$

Subtract 8 from 4 to get -4.

$$-4\left(3-2\sqrt{2}\right)$$

Use the distributive property to multiply -4 by 3-2\sqrt{2}.

$$-12+8\sqrt{2}$$