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

$$( 3 x ^ { 2 } + x ^ { 3 } ) ^ { 3 }$$

Evaluate

$$\left(x+3\right)^{3}x^{6}$$

Solution Steps

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

$$27\left(x^{2}\right)^{3}+27\left(x^{2}\right)^{2}x^{3}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

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

$$27x^{6}+27\left(x^{2}\right)^{2}x^{3}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

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

$$27x^{6}+27x^{4}x^{3}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$27x^{6}+27x^{7}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

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

$$27x^{6}+27x^{7}+9x^{2}x^{6}+\left(x^{3}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$27x^{6}+27x^{7}+9x^{8}+\left(x^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$27x^{6}+27x^{7}+9x^{8}+x^{9}$$

Expand

$$x^{9}+9x^{8}+27x^{7}+27x^{6}$$

Solution Steps

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

$$27\left(x^{2}\right)^{3}+27\left(x^{2}\right)^{2}x^{3}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

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

$$27x^{6}+27\left(x^{2}\right)^{2}x^{3}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

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

$$27x^{6}+27x^{4}x^{3}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$27x^{6}+27x^{7}+9x^{2}\left(x^{3}\right)^{2}+\left(x^{3}\right)^{3}$$

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

$$27x^{6}+27x^{7}+9x^{2}x^{6}+\left(x^{3}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$27x^{6}+27x^{7}+9x^{8}+\left(x^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$27x^{6}+27x^{7}+9x^{8}+x^{9}$$