Solve (7^{1/3}-5^{1/3})(7^frac{2{3}}+7^frac{1{3}}5^frac{1{3}}+5^frac{2{3}})=2

$$( 7 ^ { \frac { 1 } { 3 } } - 5 ^ { \frac { 1 } { 3 } } ) ( 7 ^ { \frac { 2 } { 3 } } + 7 ^ { \frac { 1 } { 3 } } 5 ^ { \frac { 1 } { 3 } } + 5 ^ { \frac { 2 } { 3 } } ) = 2$$

Evaluate (complex solution)

$$\left(\sqrt[3]{7}-\sqrt[3]{5}\right)\left(\sqrt[3]{35}+5^{\frac{2}{3}}+7^{\frac{2}{3}}\right)=2$$

Solution Steps

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 7^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+5^{\frac{2}{3}}.

$$\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}=2$$

Subtract 2 from both sides.

$$\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 7^{\frac{2}{3}}.

$$7^{\frac{1}{3}}\times 7^{\frac{2}{3}}-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

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

$$7^{1}-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Calculate 7 to the power of 1 and get 7.

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 7^{\frac{1}{3}}.

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(\left(7^{\frac{1}{3}}\right)^{2}-5^{\frac{1}{3}}\times 7^{\frac{1}{3}}\right)\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

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

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{2}{3}}-5^{\frac{1}{3}}\times 7^{\frac{1}{3}}\right)\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{2}{3}}-5^{\frac{1}{3}}\times 7^{\frac{1}{3}} by 5^{\frac{1}{3}}.

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+7^{\frac{2}{3}}\times 5^{\frac{1}{3}}-7^{\frac{1}{3}}\times \left(5^{\frac{1}{3}}\right)^{2}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

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

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+7^{\frac{2}{3}}\times 5^{\frac{1}{3}}-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Combine -5^{\frac{1}{3}}\times 7^{\frac{2}{3}} and 7^{\frac{2}{3}}\times 5^{\frac{1}{3}} to get 0.

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 5^{\frac{2}{3}}.

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{2}{3}}-5^{\frac{1}{3}}\times 5^{\frac{2}{3}}-2=0$$

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

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{2}{3}}-5^{1}-2=0$$

Calculate 5 to the power of 1 and get 5.

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{2}{3}}-5-2=0$$

Combine -7^{\frac{1}{3}}\times 5^{\frac{2}{3}} and 7^{\frac{1}{3}}\times 5^{\frac{2}{3}} to get 0.

$$7-5-2=0$$

Subtract 5 from 7 to get 2.

$$2-2=0$$

Subtract 2 from 2 to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

Verify

Solution Steps

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 7^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+5^{\frac{2}{3}}.

$$\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}=2$$

Subtract 2 from both sides.

$$\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 7^{\frac{2}{3}}.

$$7^{\frac{1}{3}}\times 7^{\frac{2}{3}}-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

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

$$7^{1}-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Calculate 7 to the power of 1 and get 7.

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 7^{\frac{1}{3}}\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 7^{\frac{1}{3}}.

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(\left(7^{\frac{1}{3}}\right)^{2}-5^{\frac{1}{3}}\times 7^{\frac{1}{3}}\right)\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

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

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+\left(7^{\frac{2}{3}}-5^{\frac{1}{3}}\times 7^{\frac{1}{3}}\right)\times 5^{\frac{1}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{2}{3}}-5^{\frac{1}{3}}\times 7^{\frac{1}{3}} by 5^{\frac{1}{3}}.

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+7^{\frac{2}{3}}\times 5^{\frac{1}{3}}-7^{\frac{1}{3}}\times \left(5^{\frac{1}{3}}\right)^{2}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

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

$$7-5^{\frac{1}{3}}\times 7^{\frac{2}{3}}+7^{\frac{2}{3}}\times 5^{\frac{1}{3}}-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Combine -5^{\frac{1}{3}}\times 7^{\frac{2}{3}} and 7^{\frac{2}{3}}\times 5^{\frac{1}{3}} to get 0.

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+\left(7^{\frac{1}{3}}-5^{\frac{1}{3}}\right)\times 5^{\frac{2}{3}}-2=0$$

Use the distributive property to multiply 7^{\frac{1}{3}}-5^{\frac{1}{3}} by 5^{\frac{2}{3}}.

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{2}{3}}-5^{\frac{1}{3}}\times 5^{\frac{2}{3}}-2=0$$

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

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{2}{3}}-5^{1}-2=0$$

Calculate 5 to the power of 1 and get 5.

$$7-7^{\frac{1}{3}}\times 5^{\frac{2}{3}}+7^{\frac{1}{3}}\times 5^{\frac{2}{3}}-5-2=0$$

Combine -7^{\frac{1}{3}}\times 5^{\frac{2}{3}} and 7^{\frac{1}{3}}\times 5^{\frac{2}{3}} to get 0.

$$7-5-2=0$$

Subtract 5 from 7 to get 2.

$$2-2=0$$

Subtract 2 from 2 to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$