Solve (2^frac{4{3}}+sqrt[3]{1/2})^3=a*(2^frac{4{3}}+sqrt[3]{1/2})+b

$$( 2 ^ { \frac { 4 } { 3 } } + \sqrt[ 3 ] { \frac { 1 } { 2 } } ) ^ { 3 } = a * ( 2 ^ { \frac { 4 } { 3 } } + \sqrt[ 3 ] { \frac { 1 } { 2 } } ) + b$$

Solve for a

$$a=-\frac{\sqrt[3]{2}\left(2b-6\times 2^{\frac{2}{3}}-24\sqrt[3]{2}-33\right)}{2\left(2\times 2^{\frac{2}{3}}+1\right)}$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2^{\frac{4}{3}}+\sqrt[3]{\frac{1}{2}}\right)^{3}.

$$\left(2^{\frac{4}{3}}\right)^{3}+3\times \left(2^{\frac{4}{3}}\right)^{2}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}=a\left(2^{\frac{4}{3}}+\sqrt[3]{\frac{1}{2}}\right)+b$$

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

$$2^{4}+3\times \left(2^{\frac{4}{3}}\right)^{2}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}=a\left(2^{\frac{4}{3}}+\sqrt[3]{\frac{1}{2}}\right)+b$$

To raise a power to another power, multiply the exponents. Multiply \frac{4}{3} and 2 to get \frac{8}{3}.

$$2^{4}+3\times 2^{\frac{8}{3}}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}=a\left(2^{\frac{4}{3}}+\sqrt[3]{\frac{1}{2}}\right)+b$$

Calculate 2 to the power of 4 and get 16.

$$16+3\times 2^{\frac{8}{3}}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}=a\left(2^{\frac{4}{3}}+\sqrt[3]{\frac{1}{2}}\right)+b$$

Use the distributive property to multiply a by 2^{\frac{4}{3}}+\sqrt[3]{\frac{1}{2}}.

$$16+3\times 2^{\frac{8}{3}}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}=a\times 2^{\frac{4}{3}}+a\sqrt[3]{\frac{1}{2}}+b$$

Swap sides so that all variable terms are on the left hand side.

$$a\times 2^{\frac{4}{3}}+a\sqrt[3]{\frac{1}{2}}+b=16+3\times 2^{\frac{8}{3}}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}$$

Subtract b from both sides.

$$a\times 2^{\frac{4}{3}}+a\sqrt[3]{\frac{1}{2}}=16+3\times 2^{\frac{8}{3}}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}-b$$

Combine all terms containing a.

$$\left(2^{\frac{4}{3}}+\sqrt[3]{\frac{1}{2}}\right)a=16+3\times 2^{\frac{8}{3}}\sqrt[3]{\frac{1}{2}}+3\times 2^{\frac{4}{3}}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}-b$$

The equation is in standard form.

$$\left(2\sqrt[3]{2}+\sqrt[3]{\frac{1}{2}}\right)a=-b+6\sqrt[3]{2}\left(\sqrt[3]{\frac{1}{2}}\right)^{2}+12\times 2^{\frac{2}{3}}\sqrt[3]{\frac{1}{2}}+\left(\sqrt[3]{\frac{1}{2}}\right)^{3}+16$$

Divide both sides by 2\times 2^{\frac{1}{3}}+\sqrt[3]{\frac{1}{2}}.

$$\frac{\left(2\sqrt[3]{2}+\sqrt[3]{\frac{1}{2}}\right)a}{2\sqrt[3]{2}+\sqrt[3]{\frac{1}{2}}}=\frac{-b+\frac{6}{\sqrt[3]{2}}+12\sqrt[3]{2}+\frac{33}{2}}{2\sqrt[3]{2}+\sqrt[3]{\frac{1}{2}}}$$

Dividing by 2\times 2^{\frac{1}{3}}+\sqrt[3]{\frac{1}{2}} undoes the multiplication by 2\times 2^{\frac{1}{3}}+\sqrt[3]{\frac{1}{2}}.

$$a=\frac{-b+\frac{6}{\sqrt[3]{2}}+12\sqrt[3]{2}+\frac{33}{2}}{2\sqrt[3]{2}+\sqrt[3]{\frac{1}{2}}}$$

Divide \frac{33}{2}+\frac{6}{\sqrt[3]{2}}+12\sqrt[3]{2}-b by 2\times 2^{\frac{1}{3}}+\sqrt[3]{\frac{1}{2}}.

$$a=\frac{\left(\frac{1}{2^{\frac{2}{3}}}+4\times 2^{\frac{2}{3}}-2\right)\left(-2\sqrt[3]{2}b+24\times 2^{\frac{2}{3}}+33\sqrt[3]{2}+12\right)}{33\sqrt[3]{2}}$$

Solve for b

$$b=\frac{2^{\frac{2}{3}}\left(-\left(2\times 2^{\frac{2}{3}}+1\right)a+12\times 2^{\frac{2}{3}}+6\right)+33}{2}$$