Solve 1/3(x^2+3)(x^2+3)(x^2+3)

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

Evaluate

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

Solution Steps

Multiply x^{2}+3 and x^{2}+3 to get \left(x^{2}+3\right)^{2}.

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

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

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

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

$$\frac{1}{3}\left(\left(x^{2}\right)^{3}+9\left(x^{2}\right)^{2}+27x^{2}+27\right)$$

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

$$\frac{1}{3}\left(x^{6}+9\left(x^{2}\right)^{2}+27x^{2}+27\right)$$

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

$$\frac{1}{3}\left(x^{6}+9x^{4}+27x^{2}+27\right)$$

Use the distributive property to multiply \frac{1}{3} by x^{6}+9x^{4}+27x^{2}+27.

$$\frac{1}{3}x^{6}+3x^{4}+9x^{2}+9$$

Expand

$$\frac{x^{6}}{3}+3x^{4}+9x^{2}+9$$

Solution Steps

Multiply x^{2}+3 and x^{2}+3 to get \left(x^{2}+3\right)^{2}.

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

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

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

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

$$\frac{1}{3}\left(\left(x^{2}\right)^{3}+9\left(x^{2}\right)^{2}+27x^{2}+27\right)$$

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

$$\frac{1}{3}\left(x^{6}+9\left(x^{2}\right)^{2}+27x^{2}+27\right)$$

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

$$\frac{1}{3}\left(x^{6}+9x^{4}+27x^{2}+27\right)$$

Use the distributive property to multiply \frac{1}{3} by x^{6}+9x^{4}+27x^{2}+27.

$$\frac{1}{3}x^{6}+3x^{4}+9x^{2}+9$$