Solve [(-2x^{3})(-3/2x^{2})]^{2}:(-3x^7)-(-2x)(+3x)^2+(-2x)^3+11x^3

$$[ ( - 2 x ^ { 3 } ) ( - \frac { 3 } { 2 } x ^ { 2 } ) ] ^ { 2 } : ( - 3 x ^ { 7 } ) - ( - 2 x ) ( + 3 x ) ^ { 2 } + ( - 2 x ) ^ { 3 } + 11 x ^ { 3 }$$

Evaluate

$$18x^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{\left(-2x^{5}\left(-\frac{3}{2}\right)\right)^{2}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Multiply -2 and -\frac{3}{2} to get 3.

$$\frac{\left(3x^{5}\right)^{2}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Expand \left(3x^{5}\right)^{2}.

$$\frac{3^{2}\left(x^{5}\right)^{2}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{3^{2}x^{10}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9x^{10}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Cancel out 3x^{7} in both numerator and denominator.

$$\frac{3x^{3}}{-1}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Anything divided by -1 gives its opposite.

$$-3x^{3}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Expand \left(3x\right)^{2}.

$$-3x^{3}-\left(-2x\times 3^{2}x^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Calculate 3 to the power of 2 and get 9.

$$-3x^{3}-\left(-2x\times 9x^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Multiply -2 and 9 to get -18.

$$-3x^{3}-\left(-18xx^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

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

$$-3x^{3}-\left(-18x^{3}\right)+\left(-2x\right)^{3}+11x^{3}$$

The opposite of -18x^{3} is 18x^{3}.

$$-3x^{3}+18x^{3}+\left(-2x\right)^{3}+11x^{3}$$

Combine -3x^{3} and 18x^{3} to get 15x^{3}.

$$15x^{3}+\left(-2x\right)^{3}+11x^{3}$$

Expand \left(-2x\right)^{3}.

$$15x^{3}+\left(-2\right)^{3}x^{3}+11x^{3}$$

Calculate -2 to the power of 3 and get -8.

$$15x^{3}-8x^{3}+11x^{3}$$

Combine 15x^{3} and -8x^{3} to get 7x^{3}.

$$7x^{3}+11x^{3}$$

Combine 7x^{3} and 11x^{3} to get 18x^{3}.

$$18x^{3}$$

Expand

$$18x^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{\left(-2x^{5}\left(-\frac{3}{2}\right)\right)^{2}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Multiply -2 and -\frac{3}{2} to get 3.

$$\frac{\left(3x^{5}\right)^{2}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Expand \left(3x^{5}\right)^{2}.

$$\frac{3^{2}\left(x^{5}\right)^{2}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{3^{2}x^{10}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9x^{10}}{-3x^{7}}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Cancel out 3x^{7} in both numerator and denominator.

$$\frac{3x^{3}}{-1}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Anything divided by -1 gives its opposite.

$$-3x^{3}-\left(-2x\times \left(3x\right)^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Expand \left(3x\right)^{2}.

$$-3x^{3}-\left(-2x\times 3^{2}x^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Calculate 3 to the power of 2 and get 9.

$$-3x^{3}-\left(-2x\times 9x^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

Multiply -2 and 9 to get -18.

$$-3x^{3}-\left(-18xx^{2}\right)+\left(-2x\right)^{3}+11x^{3}$$

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

$$-3x^{3}-\left(-18x^{3}\right)+\left(-2x\right)^{3}+11x^{3}$$

The opposite of -18x^{3} is 18x^{3}.

$$-3x^{3}+18x^{3}+\left(-2x\right)^{3}+11x^{3}$$

Combine -3x^{3} and 18x^{3} to get 15x^{3}.

$$15x^{3}+\left(-2x\right)^{3}+11x^{3}$$

Expand \left(-2x\right)^{3}.

$$15x^{3}+\left(-2\right)^{3}x^{3}+11x^{3}$$

Calculate -2 to the power of 3 and get -8.

$$15x^{3}-8x^{3}+11x^{3}$$

Combine 15x^{3} and -8x^{3} to get 7x^{3}.

$$7x^{3}+11x^{3}$$

Combine 7x^{3} and 11x^{3} to get 18x^{3}.

$$18x^{3}$$