Solve (2/3x^2+3/4x)^3=

$$( \frac { 2 } { 3 } x ^ { 2 } + \frac { 3 } { 4 } x ) ^ { 3 } =$$

Evaluate

$$\frac{\left(x\left(8x+9\right)\right)^{3}}{1728}$$

Solution Steps

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

$$\frac{8}{27}\left(x^{2}\right)^{3}+\left(x^{2}\right)^{2}x+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

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

$$\frac{8}{27}x^{6}+\left(x^{2}\right)^{2}x+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

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

$$\frac{8}{27}x^{6}+x^{4}x+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{8}{27}x^{6}+x^{5}+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{8}{27}x^{6}+x^{5}+\frac{9}{8}x^{4}+\frac{27}{64}x^{3}$$

Expand

$$\frac{8x^{6}}{27}+x^{5}+\frac{9x^{4}}{8}+\frac{27x^{3}}{64}$$

Solution Steps

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

$$\frac{8}{27}\left(x^{2}\right)^{3}+\left(x^{2}\right)^{2}x+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

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

$$\frac{8}{27}x^{6}+\left(x^{2}\right)^{2}x+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

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

$$\frac{8}{27}x^{6}+x^{4}x+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{8}{27}x^{6}+x^{5}+\frac{9}{8}x^{2}x^{2}+\frac{27}{64}x^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{8}{27}x^{6}+x^{5}+\frac{9}{8}x^{4}+\frac{27}{64}x^{3}$$