Solve (1-3/5x^3)^2+(1/2x^2+2)^2

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

Evaluate

$$\frac{\left(5-3x^{3}\right)^{2}}{25}+\frac{\left(x^{2}+4\right)^{2}}{4}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\frac{3}{5}x^{3}\right)^{2}.

$$1-\frac{6}{5}x^{3}+\frac{9}{25}\left(x^{3}\right)^{2}+\left(\frac{1}{2}x^{2}+2\right)^{2}$$

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

$$1-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\left(\frac{1}{2}x^{2}+2\right)^{2}$$

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

$$1-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\frac{1}{4}\left(x^{2}\right)^{2}+2x^{2}+4$$

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

$$1-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\frac{1}{4}x^{4}+2x^{2}+4$$

Add 1 and 4 to get 5.

$$5-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\frac{1}{4}x^{4}+2x^{2}$$

Expand

$$\frac{9x^{6}}{25}+\frac{x^{4}}{4}-\frac{6x^{3}}{5}+2x^{2}+5$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\frac{3}{5}x^{3}\right)^{2}.

$$1-\frac{6}{5}x^{3}+\frac{9}{25}\left(x^{3}\right)^{2}+\left(\frac{1}{2}x^{2}+2\right)^{2}$$

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

$$1-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\left(\frac{1}{2}x^{2}+2\right)^{2}$$

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

$$1-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\frac{1}{4}\left(x^{2}\right)^{2}+2x^{2}+4$$

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

$$1-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\frac{1}{4}x^{4}+2x^{2}+4$$

Add 1 and 4 to get 5.

$$5-\frac{6}{5}x^{3}+\frac{9}{25}x^{6}+\frac{1}{4}x^{4}+2x^{2}$$