Solve (a^{2}+b^{2})(a^{4}-a^{2}b^{2}+b^{4})+(a^{3}-b^3)(a^3+b^3)=2a^6

$$( a ^ { 2 } + b ^ { 2 } ) ( a ^ { 4 } - a ^ { 2 } b ^ { 2 } + b ^ { 4 } ) + ( a ^ { 3 } - b ^ { 3 } ) ( a ^ { 3 } + b ^ { 3 } ) = 2 a ^ { 6 }$$

Solve for a

$$a\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply a^{2}+b^{2} by a^{4}-a^{2}b^{2}+b^{4}.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)=2a^{6}$$

Consider \left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+\left(a^{3}\right)^{2}-\left(b^{3}\right)^{2}=2a^{6}$$

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

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+a^{6}-\left(b^{3}\right)^{2}=2a^{6}$$

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

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+a^{6}-b^{6}=2a^{6}$$

Combine b^{6} and -b^{6} to get 0.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{6}=2a^{6}$$

Subtract 2a^{6} from both sides.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{6}-2a^{6}=0$$

Combine a^{6} and -2a^{6} to get -a^{6}.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)-a^{6}=0$$

Use the distributive property to multiply a^{2} by a^{4}-a^{2}b^{2}.

$$a^{6}-b^{2}a^{4}+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)-a^{6}=0$$

Use the distributive property to multiply b^{2} by a^{4}-a^{2}b^{2}.

$$a^{6}-b^{2}a^{4}+a^{2}b^{4}+b^{2}a^{4}-a^{2}b^{4}-a^{6}=0$$

Combine -b^{2}a^{4} and b^{2}a^{4} to get 0.

$$a^{6}+a^{2}b^{4}-a^{2}b^{4}-a^{6}=0$$

Combine a^{2}b^{4} and -a^{2}b^{4} to get 0.

$$a^{6}-a^{6}=0$$

Combine a^{6} and -a^{6} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Solve for b

$$b\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply a^{2}+b^{2} by a^{4}-a^{2}b^{2}+b^{4}.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)=2a^{6}$$

Consider \left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+\left(a^{3}\right)^{2}-\left(b^{3}\right)^{2}=2a^{6}$$

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

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+a^{6}-\left(b^{3}\right)^{2}=2a^{6}$$

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

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+b^{6}+a^{6}-b^{6}=2a^{6}$$

Combine b^{6} and -b^{6} to get 0.

$$a^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{6}=2a^{6}$$

Use the distributive property to multiply a^{2} by a^{4}-a^{2}b^{2}.

$$a^{6}-b^{2}a^{4}+a^{2}b^{4}+b^{2}\left(a^{4}-a^{2}b^{2}\right)+a^{6}=2a^{6}$$

Use the distributive property to multiply b^{2} by a^{4}-a^{2}b^{2}.

$$a^{6}-b^{2}a^{4}+a^{2}b^{4}+b^{2}a^{4}-a^{2}b^{4}+a^{6}=2a^{6}$$

Combine -b^{2}a^{4} and b^{2}a^{4} to get 0.

$$a^{6}+a^{2}b^{4}-a^{2}b^{4}+a^{6}=2a^{6}$$

Combine a^{2}b^{4} and -a^{2}b^{4} to get 0.

$$a^{6}+a^{6}=2a^{6}$$

Combine a^{6} and a^{6} to get 2a^{6}.

$$2a^{6}=2a^{6}$$

Cancel out 2 on both sides.

$$a^{6}=a^{6}$$

Reorder the terms.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{R}$$