Solve (2^frac{1{4}}-1)(2^frac{3{4}}+2^frac{1{2}}+2^frac{1{4}}+1)

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

Evaluate

$$1$$

Solution Steps

Use the distributive property to multiply 2^{\frac{1}{4}}-1 by 2^{\frac{3}{4}}+2^{\frac{1}{2}}+2^{\frac{1}{4}}+1 and combine like terms.

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

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

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

To multiply powers of the same base, add their exponents. Add \frac{1}{4} and \frac{1}{2} to get \frac{3}{4}.

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

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

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

Calculate 2 to the power of 1 and get 2.

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

Combine 2^{\frac{3}{4}} and -2^{\frac{3}{4}} to get 0.

$$2+2^{\frac{1}{2}}-2^{\frac{1}{2}}-1$$

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

$$2-1$$

Subtract 1 from 2 to get 1.

$$1$$

Factor

$$1$$