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

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

Evaluate

$$\frac{27}{4}=6.75$$

Solution Steps

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

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

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

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

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

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

Multiply 2^{\frac{2}{3}} and 2^{-\frac{2}{3}} to get 1.

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

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

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

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

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

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

$$2^{1}+3+3\times 2^{-1}+2^{-2}$$

Calculate 2 to the power of 1 and get 2.

$$2+3+3\times 2^{-1}+2^{-2}$$

Add 2 and 3 to get 5.

$$5+3\times 2^{-1}+2^{-2}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$5+3\times \frac{1}{2}+2^{-2}$$

Multiply 3 and \frac{1}{2} to get \frac{3}{2}.

$$5+\frac{3}{2}+2^{-2}$$

Add 5 and \frac{3}{2} to get \frac{13}{2}.

$$\frac{13}{2}+2^{-2}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{13}{2}+\frac{1}{4}$$

Add \frac{13}{2} and \frac{1}{4} to get \frac{27}{4}.

$$\frac{27}{4}$$

Factor

$$\frac{3 ^ {3}}{2 ^ {2}} = 6\frac{3}{4} = 6.75$$