Solve left(x+x^2right)^3

$${ \left(x+ { x }^{ 2 } \right) }^{ 3 }$$

Evaluate

$$\left(x\left(x+1\right)\right)^{3}$$

Solution Steps

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

$$x^{3}+3x^{2}x^{2}+3x\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{3}+3x^{4}+3x\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{3}+3x^{4}+3xx^{4}+\left(x^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$x^{3}+3x^{4}+3x^{5}+\left(x^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$x^{3}+3x^{4}+3x^{5}+x^{6}$$

Expand

$$x^{6}+3x^{5}+3x^{4}+x^{3}$$

Solution Steps

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

$$x^{3}+3x^{2}x^{2}+3x\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{3}+3x^{4}+3x\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{3}+3x^{4}+3xx^{4}+\left(x^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$x^{3}+3x^{4}+3x^{5}+\left(x^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$x^{3}+3x^{4}+3x^{5}+x^{6}$$