Solve (5/2x^3-3/4)*{left(1/3x^2+1/4xright)}^2+3x^3*{left(frac{2}{3}x^2-frac{3}{2}xright)}^2

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

Evaluate

$$\frac{29x^{7}}{18}-\frac{67x^{6}}{12}+\frac{221x^{5}}{32}-\frac{\left(4\sqrt{3}x^{2}+3\sqrt{3}x\right)^{2}}{576}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{3}x^{2}+\frac{1}{4}x\right)^{2}.

$$\left(\frac{5}{2}x^{3}-\frac{3}{4}\right)\left(\frac{1}{9}\left(x^{2}\right)^{2}+\frac{1}{6}x^{2}x+\frac{1}{16}x^{2}\right)+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

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

$$\left(\frac{5}{2}x^{3}-\frac{3}{4}\right)\left(\frac{1}{9}x^{4}+\frac{1}{6}x^{2}x+\frac{1}{16}x^{2}\right)+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

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

$$\left(\frac{5}{2}x^{3}-\frac{3}{4}\right)\left(\frac{1}{9}x^{4}+\frac{1}{6}x^{3}+\frac{1}{16}x^{2}\right)+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

Use the distributive property to multiply \frac{5}{2}x^{3}-\frac{3}{4} by \frac{1}{9}x^{4}+\frac{1}{6}x^{3}+\frac{1}{16}x^{2}.

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{4}{9}\left(x^{2}\right)^{2}-2x^{2}x+\frac{9}{4}x^{2}\right)$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{4}{9}x^{4}-2x^{2}x+\frac{9}{4}x^{2}\right)$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{4}{9}x^{4}-2x^{3}+\frac{9}{4}x^{2}\right)$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+\frac{4}{3}x^{7}-6x^{6}+\frac{27}{4}x^{5}$$

Combine \frac{5}{18}x^{7} and \frac{4}{3}x^{7} to get \frac{29}{18}x^{7}.

$$\frac{29}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}-6x^{6}+\frac{27}{4}x^{5}$$

Combine \frac{5}{12}x^{6} and -6x^{6} to get -\frac{67}{12}x^{6}.

$$\frac{29}{18}x^{7}-\frac{67}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+\frac{27}{4}x^{5}$$

Combine \frac{5}{32}x^{5} and \frac{27}{4}x^{5} to get \frac{221}{32}x^{5}.

$$\frac{29}{18}x^{7}-\frac{67}{12}x^{6}+\frac{221}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}$$

Expand

$$\frac{29x^{7}}{18}-\frac{67x^{6}}{12}+\frac{221x^{5}}{32}-\frac{x^{4}}{12}-\frac{x^{3}}{8}-\frac{3x^{2}}{64}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{3}x^{2}+\frac{1}{4}x\right)^{2}.

$$\left(\frac{5}{2}x^{3}-\frac{3}{4}\right)\left(\frac{1}{9}\left(x^{2}\right)^{2}+\frac{1}{6}x^{2}x+\frac{1}{16}x^{2}\right)+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

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

$$\left(\frac{5}{2}x^{3}-\frac{3}{4}\right)\left(\frac{1}{9}x^{4}+\frac{1}{6}x^{2}x+\frac{1}{16}x^{2}\right)+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

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

$$\left(\frac{5}{2}x^{3}-\frac{3}{4}\right)\left(\frac{1}{9}x^{4}+\frac{1}{6}x^{3}+\frac{1}{16}x^{2}\right)+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

Use the distributive property to multiply \frac{5}{2}x^{3}-\frac{3}{4} by \frac{1}{9}x^{4}+\frac{1}{6}x^{3}+\frac{1}{16}x^{2}.

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{2}{3}x^{2}-\frac{3}{2}x\right)^{2}$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{4}{9}\left(x^{2}\right)^{2}-2x^{2}x+\frac{9}{4}x^{2}\right)$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{4}{9}x^{4}-2x^{2}x+\frac{9}{4}x^{2}\right)$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+3x^{3}\left(\frac{4}{9}x^{4}-2x^{3}+\frac{9}{4}x^{2}\right)$$

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

$$\frac{5}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+\frac{4}{3}x^{7}-6x^{6}+\frac{27}{4}x^{5}$$

Combine \frac{5}{18}x^{7} and \frac{4}{3}x^{7} to get \frac{29}{18}x^{7}.

$$\frac{29}{18}x^{7}+\frac{5}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}-6x^{6}+\frac{27}{4}x^{5}$$

Combine \frac{5}{12}x^{6} and -6x^{6} to get -\frac{67}{12}x^{6}.

$$\frac{29}{18}x^{7}-\frac{67}{12}x^{6}+\frac{5}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}+\frac{27}{4}x^{5}$$

Combine \frac{5}{32}x^{5} and \frac{27}{4}x^{5} to get \frac{221}{32}x^{5}.

$$\frac{29}{18}x^{7}-\frac{67}{12}x^{6}+\frac{221}{32}x^{5}-\frac{1}{12}x^{4}-\frac{1}{8}x^{3}-\frac{3}{64}x^{2}$$