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

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

Evaluate

$$\frac{\left(2x^{2}+1\right)^{3}}{8}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(\frac{1}{2}+x^{2}\right)^{3}.

$$\frac{1}{8}+\frac{3}{4}x^{2}+\frac{3}{2}\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}$$

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

$$\frac{1}{8}+\frac{3}{4}x^{2}+\frac{3}{2}x^{4}+\left(x^{2}\right)^{3}$$

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

$$\frac{1}{8}+\frac{3}{4}x^{2}+\frac{3}{2}x^{4}+x^{6}$$

Expand

$$x^{6}+\frac{3x^{4}}{2}+\frac{3x^{2}}{4}+\frac{1}{8}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(\frac{1}{2}+x^{2}\right)^{3}.

$$\frac{1}{8}+\frac{3}{4}x^{2}+\frac{3}{2}\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}$$

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

$$\frac{1}{8}+\frac{3}{4}x^{2}+\frac{3}{2}x^{4}+\left(x^{2}\right)^{3}$$

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

$$\frac{1}{8}+\frac{3}{4}x^{2}+\frac{3}{2}x^{4}+x^{6}$$