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

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

Evaluate

$$a^{2}+b^{2}$$

Solution Steps

Use the distributive property to multiply a^{\frac{2}{3}}+b^{\frac{2}{3}} by a^{\frac{4}{3}}-a^{\frac{2}{3}}b^{\frac{2}{3}}+b^{\frac{4}{3}}.

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

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{4}{3} to get 2.

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

Use the distributive property to multiply a^{\frac{2}{3}} by a^{\frac{4}{3}}-a^{\frac{2}{3}}b^{\frac{2}{3}}.

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

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{4}{3} to get 2.

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

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and 2 to get \frac{4}{3}.

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

Use the distributive property to multiply b^{\frac{2}{3}} by a^{\frac{4}{3}}-a^{\frac{2}{3}}b^{\frac{2}{3}}.

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

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and 2 to get \frac{4}{3}.

$$a^{2}-b^{\frac{2}{3}}a^{\frac{4}{3}}+a^{\frac{2}{3}}b^{\frac{4}{3}}+b^{\frac{2}{3}}a^{\frac{4}{3}}-a^{\frac{2}{3}}b^{\frac{4}{3}}+b^{2}$$

Combine -b^{\frac{2}{3}}a^{\frac{4}{3}} and b^{\frac{2}{3}}a^{\frac{4}{3}} to get 0.

$$a^{2}+a^{\frac{2}{3}}b^{\frac{4}{3}}-a^{\frac{2}{3}}b^{\frac{4}{3}}+b^{2}$$

Combine a^{\frac{2}{3}}b^{\frac{4}{3}} and -a^{\frac{2}{3}}b^{\frac{4}{3}} to get 0.

$$a^{2}+b^{2}$$

Differentiate w.r.t. a

$$2a$$