Solve {left(x^3-3sqrt{2}x^2+6x-2sqrt{2}right)}^2

$${ \left( { x }^{ 3 } -3 \sqrt{ 2 } { x }^{ 2 } +6x-2 \sqrt{ 2 } \right) }^{ 2 }$$

Expand

$$x^{6}-6\sqrt{2}x^{5}+30x^{4}-40\sqrt{2}x^{3}+60x^{2}-24\sqrt{2}x+8$$

Solution Steps

Square x^{3}-3\sqrt{2}x^{2}+6x-2\sqrt{2}.

$$\left(x^{3}-3\sqrt{2}x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\left(x^{3}-3\sqrt{2}x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+4\times 2$$

Multiply 4 and 2 to get 8.

$$\left(x^{3}-3\sqrt{2}x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

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

$$\left(x^{3}\right)^{2}-6x^{3}\sqrt{2}x^{2}+9\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{6}-6x^{3}\sqrt{2}x^{2}+9\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$x^{6}-6x^{5}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{6}-6x^{5}\sqrt{2}+9\left(\sqrt{2}\right)^{2}x^{4}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

The square of \sqrt{2} is 2.

$$x^{6}-6x^{5}\sqrt{2}+9\times 2x^{4}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

Multiply 9 and 2 to get 18.

$$x^{6}-6x^{5}\sqrt{2}+18x^{4}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

Use the distributive property to multiply 12x by x^{3}-3\sqrt{2}x^{2}.

$$x^{6}-6x^{5}\sqrt{2}+18x^{4}+12x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

Combine 18x^{4} and 12x^{4} to get 30x^{4}.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

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

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}x^{3}+12x^{2}\left(\sqrt{2}\right)^{2}+36x^{2}-24\sqrt{2}x+8$$

The square of \sqrt{2} is 2.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}x^{3}+12x^{2}\times 2+36x^{2}-24\sqrt{2}x+8$$

Multiply 12 and 2 to get 24.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}x^{3}+24x^{2}+36x^{2}-24\sqrt{2}x+8$$

Combine -36\sqrt{2}x^{3} and -4\sqrt{2}x^{3} to get -40\sqrt{2}x^{3}.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-40\sqrt{2}x^{3}+24x^{2}+36x^{2}-24\sqrt{2}x+8$$

Combine 24x^{2} and 36x^{2} to get 60x^{2}.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-40\sqrt{2}x^{3}+60x^{2}-24\sqrt{2}x+8$$

Evaluate

$$\left(x^{3}-3\sqrt{2}x^{2}+6x-2\sqrt{2}\right)^{2}$$

Solution Steps

Square x^{3}-3\sqrt{2}x^{2}+6x-2\sqrt{2}.

$$\left(x^{3}-3\sqrt{2}x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\left(x^{3}-3\sqrt{2}x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+4\times 2$$

Multiply 4 and 2 to get 8.

$$\left(x^{3}-3\sqrt{2}x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

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

$$\left(x^{3}\right)^{2}-6x^{3}\sqrt{2}x^{2}+9\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{6}-6x^{3}\sqrt{2}x^{2}+9\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$x^{6}-6x^{5}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{6}-6x^{5}\sqrt{2}+9\left(\sqrt{2}\right)^{2}x^{4}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

The square of \sqrt{2} is 2.

$$x^{6}-6x^{5}\sqrt{2}+9\times 2x^{4}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

Multiply 9 and 2 to get 18.

$$x^{6}-6x^{5}\sqrt{2}+18x^{4}+12x\left(x^{3}-3\sqrt{2}x^{2}\right)-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

Use the distributive property to multiply 12x by x^{3}-3\sqrt{2}x^{2}.

$$x^{6}-6x^{5}\sqrt{2}+18x^{4}+12x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

Combine 18x^{4} and 12x^{4} to get 30x^{4}.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}\left(x^{3}-3\sqrt{2}x^{2}\right)+36x^{2}-24\sqrt{2}x+8$$

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

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}x^{3}+12x^{2}\left(\sqrt{2}\right)^{2}+36x^{2}-24\sqrt{2}x+8$$

The square of \sqrt{2} is 2.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}x^{3}+12x^{2}\times 2+36x^{2}-24\sqrt{2}x+8$$

Multiply 12 and 2 to get 24.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-36\sqrt{2}x^{3}-4\sqrt{2}x^{3}+24x^{2}+36x^{2}-24\sqrt{2}x+8$$

Combine -36\sqrt{2}x^{3} and -4\sqrt{2}x^{3} to get -40\sqrt{2}x^{3}.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-40\sqrt{2}x^{3}+24x^{2}+36x^{2}-24\sqrt{2}x+8$$

Combine 24x^{2} and 36x^{2} to get 60x^{2}.

$$x^{6}-6x^{5}\sqrt{2}+30x^{4}-40\sqrt{2}x^{3}+60x^{2}-24\sqrt{2}x+8$$