Solve [1-sqrt{2}[1-sqrt{2}](1-sqrt{2})]

$$[ 1 - \sqrt { 2 } [ 1 - \sqrt { 2 } ] ( 1 - \sqrt { 2 } ) ]$$

Evaluate

$$5-3\sqrt{2}\approx 0.757359313$$

Solution Steps

Multiply 1-\sqrt{2} and 1-\sqrt{2} to get \left(1-\sqrt{2}\right)^{2}.

$$1-\sqrt{2}\left(1-\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{2}\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 1 and 2 to get 3.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply -2 and 2 to get -4.

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

To find the opposite of 3\sqrt{2}-4, find the opposite of each term.

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

The opposite of -4 is 4.

$$1-3\sqrt{2}+4$$

Add 1 and 4 to get 5.

$$5-3\sqrt{2}$$