Solve (x^{1/3}-y^{1/3})(x^frac{2{3}}+x^frac{1{3}}y^frac{1{3}}+y^frac{2{3}})

$$( x ^ { \frac { 1 } { 3 } } - y ^ { \frac { 1 } { 3 } } ) ( x ^ { \frac { 2 } { 3 } } + x ^ { \frac { 1 } { 3 } } y ^ { \frac { 1 } { 3 } } + y ^ { \frac { 2 } { 3 } } )$$

Evaluate

$$x-y$$

Solution Steps

Use the distributive property to multiply x^{\frac{1}{3}}-y^{\frac{1}{3}} by x^{\frac{2}{3}}+x^{\frac{1}{3}}y^{\frac{1}{3}}+y^{\frac{2}{3}}.

$$x^{\frac{1}{3}}x^{\frac{2}{3}}+y^{\frac{1}{3}}\left(x^{\frac{1}{3}}\right)^{2}+x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{\frac{1}{3}}x^{\frac{2}{3}}-x^{\frac{1}{3}}\left(y^{\frac{1}{3}}\right)^{2}-y^{\frac{1}{3}}y^{\frac{2}{3}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and \frac{2}{3} to get 1.

$$x^{1}+y^{\frac{1}{3}}\left(x^{\frac{1}{3}}\right)^{2}+x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{\frac{1}{3}}x^{\frac{2}{3}}-x^{\frac{1}{3}}\left(y^{\frac{1}{3}}\right)^{2}-y^{\frac{1}{3}}y^{\frac{2}{3}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$x^{1}+y^{\frac{1}{3}}x^{\frac{2}{3}}+x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{\frac{1}{3}}x^{\frac{2}{3}}-x^{\frac{1}{3}}\left(y^{\frac{1}{3}}\right)^{2}-y^{\frac{1}{3}}y^{\frac{2}{3}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$x^{1}+y^{\frac{1}{3}}x^{\frac{2}{3}}+x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{\frac{1}{3}}x^{\frac{2}{3}}-x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{\frac{1}{3}}y^{\frac{2}{3}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and \frac{2}{3} to get 1.

$$x^{1}+y^{\frac{1}{3}}x^{\frac{2}{3}}+x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{\frac{1}{3}}x^{\frac{2}{3}}-x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{1}$$

Calculate x to the power of 1 and get x.

$$x+y^{\frac{1}{3}}x^{\frac{2}{3}}+x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{\frac{1}{3}}x^{\frac{2}{3}}-x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{1}$$

Combine y^{\frac{1}{3}}x^{\frac{2}{3}} and -y^{\frac{1}{3}}x^{\frac{2}{3}} to get 0.

$$x+x^{\frac{1}{3}}y^{\frac{2}{3}}-x^{\frac{1}{3}}y^{\frac{2}{3}}-y^{1}$$

Combine x^{\frac{1}{3}}y^{\frac{2}{3}} and -x^{\frac{1}{3}}y^{\frac{2}{3}} to get 0.

$$x-y^{1}$$

Calculate y to the power of 1 and get y.

$$x-y$$

Differentiate w.r.t. x

$$1$$