Solve (d^{1/4}+z^{1/4})*(d^frac{1{8}}+z^frac{1{8}})*(d^frac{1{8}}-z^frac{1{8}})

$$( d ^ { \frac { 1 } { 4 } } + z ^ { \frac { 1 } { 4 } } ) \cdot ( d ^ { \frac { 1 } { 8 } } + z ^ { \frac { 1 } { 8 } } ) \cdot ( d ^ { \frac { 1 } { 8 } } - z ^ { \frac { 1 } { 8 } } )$$

Evaluate

$$\sqrt{d}-\sqrt{z}$$

Solution Steps

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

$$\left(d^{\frac{1}{4}}d^{\frac{1}{8}}+d^{\frac{1}{4}}z^{\frac{1}{8}}+z^{\frac{1}{4}}d^{\frac{1}{8}}+z^{\frac{1}{4}}z^{\frac{1}{8}}\right)\left(d^{\frac{1}{8}}-z^{\frac{1}{8}}\right)$$

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

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

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

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

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

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

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

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

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

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

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

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

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

$$d^{\frac{1}{2}}-d^{\frac{3}{8}}z^{\frac{1}{8}}+d^{\frac{3}{8}}z^{\frac{1}{8}}-d^{\frac{1}{4}}z^{\frac{1}{4}}+z^{\frac{1}{4}}d^{\frac{1}{4}}-z^{\frac{1}{4}}d^{\frac{1}{8}}z^{\frac{1}{8}}+z^{\frac{3}{8}}d^{\frac{1}{8}}-z^{\frac{3}{8}}z^{\frac{1}{8}}$$

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

$$d^{\frac{1}{2}}-d^{\frac{3}{8}}z^{\frac{1}{8}}+d^{\frac{3}{8}}z^{\frac{1}{8}}-d^{\frac{1}{4}}z^{\frac{1}{4}}+z^{\frac{1}{4}}d^{\frac{1}{4}}-z^{\frac{3}{8}}d^{\frac{1}{8}}+z^{\frac{3}{8}}d^{\frac{1}{8}}-z^{\frac{3}{8}}z^{\frac{1}{8}}$$

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

$$d^{\frac{1}{2}}-d^{\frac{3}{8}}z^{\frac{1}{8}}+d^{\frac{3}{8}}z^{\frac{1}{8}}-d^{\frac{1}{4}}z^{\frac{1}{4}}+z^{\frac{1}{4}}d^{\frac{1}{4}}-z^{\frac{3}{8}}d^{\frac{1}{8}}+z^{\frac{3}{8}}d^{\frac{1}{8}}-z^{\frac{1}{2}}$$

Combine -d^{\frac{3}{8}}z^{\frac{1}{8}} and d^{\frac{3}{8}}z^{\frac{1}{8}} to get 0.

$$d^{\frac{1}{2}}-d^{\frac{1}{4}}z^{\frac{1}{4}}+z^{\frac{1}{4}}d^{\frac{1}{4}}-z^{\frac{3}{8}}d^{\frac{1}{8}}+z^{\frac{3}{8}}d^{\frac{1}{8}}-z^{\frac{1}{2}}$$

Combine -d^{\frac{1}{4}}z^{\frac{1}{4}} and z^{\frac{1}{4}}d^{\frac{1}{4}} to get 0.

$$d^{\frac{1}{2}}-z^{\frac{3}{8}}d^{\frac{1}{8}}+z^{\frac{3}{8}}d^{\frac{1}{8}}-z^{\frac{1}{2}}$$

Combine -z^{\frac{3}{8}}d^{\frac{1}{8}} and z^{\frac{3}{8}}d^{\frac{1}{8}} to get 0.

$$d^{\frac{1}{2}}-z^{\frac{1}{2}}$$

Differentiate w.r.t. d

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