Solve (x^{1/4}+y^{1/4})*(x^frac{1{8}}+y^frac{1{8}})*(x^frac{1{8}}-y^frac{1{8}})

$$( x ^ { \frac { 1 } { 4 } } + y ^ { \frac { 1 } { 4 } } ) \cdot ( x ^ { \frac { 1 } { 8 } } + y ^ { \frac { 1 } { 8 } } ) \cdot ( x ^ { \frac { 1 } { 8 } } - y ^ { \frac { 1 } { 8 } } )$$

Evaluate

$$\sqrt{x}-\sqrt{y}$$

Solution Steps

Use the distributive property to multiply x^{\frac{1}{4}}+y^{\frac{1}{4}} by x^{\frac{1}{8}}+y^{\frac{1}{8}}.

$$\left(x^{\frac{1}{4}}x^{\frac{1}{8}}+x^{\frac{1}{4}}y^{\frac{1}{8}}+y^{\frac{1}{4}}x^{\frac{1}{8}}+y^{\frac{1}{4}}y^{\frac{1}{8}}\right)\left(x^{\frac{1}{8}}-y^{\frac{1}{8}}\right)$$

To multiply powers of the same base, add their exponents. Add \frac{1}{4} and \frac{1}{8} to get \frac{3}{8}.

$$\left(x^{\frac{3}{8}}+x^{\frac{1}{4}}y^{\frac{1}{8}}+y^{\frac{1}{4}}x^{\frac{1}{8}}+y^{\frac{1}{4}}y^{\frac{1}{8}}\right)\left(x^{\frac{1}{8}}-y^{\frac{1}{8}}\right)$$

To multiply powers of the same base, add their exponents. Add \frac{1}{4} and \frac{1}{8} to get \frac{3}{8}.

$$\left(x^{\frac{3}{8}}+x^{\frac{1}{4}}y^{\frac{1}{8}}+y^{\frac{1}{4}}x^{\frac{1}{8}}+y^{\frac{3}{8}}\right)\left(x^{\frac{1}{8}}-y^{\frac{1}{8}}\right)$$

Use the distributive property to multiply x^{\frac{3}{8}}+x^{\frac{1}{4}}y^{\frac{1}{8}}+y^{\frac{1}{4}}x^{\frac{1}{8}}+y^{\frac{3}{8}} by x^{\frac{1}{8}}-y^{\frac{1}{8}}.

$$x^{\frac{3}{8}}x^{\frac{1}{8}}-x^{\frac{3}{8}}y^{\frac{1}{8}}+x^{\frac{1}{4}}y^{\frac{1}{8}}x^{\frac{1}{8}}-x^{\frac{1}{4}}\left(y^{\frac{1}{8}}\right)^{2}+y^{\frac{1}{4}}\left(x^{\frac{1}{8}}\right)^{2}-y^{\frac{1}{4}}x^{\frac{1}{8}}y^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{3}{8}}y^{\frac{1}{8}}$$

To multiply powers of the same base, add their exponents. Add \frac{3}{8} and \frac{1}{8} to get \frac{1}{2}.

$$x^{\frac{1}{2}}-x^{\frac{3}{8}}y^{\frac{1}{8}}+x^{\frac{1}{4}}y^{\frac{1}{8}}x^{\frac{1}{8}}-x^{\frac{1}{4}}\left(y^{\frac{1}{8}}\right)^{2}+y^{\frac{1}{4}}\left(x^{\frac{1}{8}}\right)^{2}-y^{\frac{1}{4}}x^{\frac{1}{8}}y^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{3}{8}}y^{\frac{1}{8}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{4} and \frac{1}{8} to get \frac{3}{8}.

$$x^{\frac{1}{2}}-x^{\frac{3}{8}}y^{\frac{1}{8}}+x^{\frac{3}{8}}y^{\frac{1}{8}}-x^{\frac{1}{4}}\left(y^{\frac{1}{8}}\right)^{2}+y^{\frac{1}{4}}\left(x^{\frac{1}{8}}\right)^{2}-y^{\frac{1}{4}}x^{\frac{1}{8}}y^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{3}{8}}y^{\frac{1}{8}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{8} and 2 to get \frac{1}{4}.

$$x^{\frac{1}{2}}-x^{\frac{3}{8}}y^{\frac{1}{8}}+x^{\frac{3}{8}}y^{\frac{1}{8}}-x^{\frac{1}{4}}y^{\frac{1}{4}}+y^{\frac{1}{4}}\left(x^{\frac{1}{8}}\right)^{2}-y^{\frac{1}{4}}x^{\frac{1}{8}}y^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{3}{8}}y^{\frac{1}{8}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{8} and 2 to get \frac{1}{4}.

$$x^{\frac{1}{2}}-x^{\frac{3}{8}}y^{\frac{1}{8}}+x^{\frac{3}{8}}y^{\frac{1}{8}}-x^{\frac{1}{4}}y^{\frac{1}{4}}+y^{\frac{1}{4}}x^{\frac{1}{4}}-y^{\frac{1}{4}}x^{\frac{1}{8}}y^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{3}{8}}y^{\frac{1}{8}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{4} and \frac{1}{8} to get \frac{3}{8}.

$$x^{\frac{1}{2}}-x^{\frac{3}{8}}y^{\frac{1}{8}}+x^{\frac{3}{8}}y^{\frac{1}{8}}-x^{\frac{1}{4}}y^{\frac{1}{4}}+y^{\frac{1}{4}}x^{\frac{1}{4}}-y^{\frac{3}{8}}x^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{3}{8}}y^{\frac{1}{8}}$$

To multiply powers of the same base, add their exponents. Add \frac{3}{8} and \frac{1}{8} to get \frac{1}{2}.

$$x^{\frac{1}{2}}-x^{\frac{3}{8}}y^{\frac{1}{8}}+x^{\frac{3}{8}}y^{\frac{1}{8}}-x^{\frac{1}{4}}y^{\frac{1}{4}}+y^{\frac{1}{4}}x^{\frac{1}{4}}-y^{\frac{3}{8}}x^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{1}{2}}$$

Combine -x^{\frac{3}{8}}y^{\frac{1}{8}} and x^{\frac{3}{8}}y^{\frac{1}{8}} to get 0.

$$x^{\frac{1}{2}}-x^{\frac{1}{4}}y^{\frac{1}{4}}+y^{\frac{1}{4}}x^{\frac{1}{4}}-y^{\frac{3}{8}}x^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{1}{2}}$$

Combine -x^{\frac{1}{4}}y^{\frac{1}{4}} and y^{\frac{1}{4}}x^{\frac{1}{4}} to get 0.

$$x^{\frac{1}{2}}-y^{\frac{3}{8}}x^{\frac{1}{8}}+y^{\frac{3}{8}}x^{\frac{1}{8}}-y^{\frac{1}{2}}$$

Combine -y^{\frac{3}{8}}x^{\frac{1}{8}} and y^{\frac{3}{8}}x^{\frac{1}{8}} to get 0.

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

Differentiate w.r.t. x

$$\frac{1}{2\sqrt{x}}$$