Solve (1/2x^2-4x^-1)^3

$$( \frac { 1 } { 2 } x ^ { 2 } - 4 x ^ { - 1 } ) ^ { 3 }$$

Evaluate

$$\frac{\left(\frac{x^{3}-8}{x}\right)^{3}}{8}$$

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}-4x^{-1}\right)^{3}.

$$\frac{1}{8}\left(x^{2}\right)^{3}-3\left(x^{2}\right)^{2}x^{-1}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

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

$$\frac{1}{8}x^{6}-3\left(x^{2}\right)^{2}x^{-1}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

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

$$\frac{1}{8}x^{6}-3x^{4}x^{-1}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and -1 to get 3.

$$\frac{1}{8}x^{6}-3x^{3}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{1}{8}x^{6}-3x^{3}+24x^{2}x^{-2}-64\left(x^{-1}\right)^{3}$$

Multiply x^{2} and x^{-2} to get 1.

$$\frac{1}{8}x^{6}-3x^{3}+24-64\left(x^{-1}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 3 to get -3.

$$\frac{1}{8}x^{6}-3x^{3}+24-64x^{-3}$$

Expand

$$\frac{x^{6}}{8}-3x^{3}+24-\frac{64}{x^{3}}$$

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}-4x^{-1}\right)^{3}.

$$\frac{1}{8}\left(x^{2}\right)^{3}-3\left(x^{2}\right)^{2}x^{-1}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

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

$$\frac{1}{8}x^{6}-3\left(x^{2}\right)^{2}x^{-1}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

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

$$\frac{1}{8}x^{6}-3x^{4}x^{-1}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and -1 to get 3.

$$\frac{1}{8}x^{6}-3x^{3}+24x^{2}\left(x^{-1}\right)^{2}-64\left(x^{-1}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{1}{8}x^{6}-3x^{3}+24x^{2}x^{-2}-64\left(x^{-1}\right)^{3}$$

Multiply x^{2} and x^{-2} to get 1.

$$\frac{1}{8}x^{6}-3x^{3}+24-64\left(x^{-1}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 3 to get -3.

$$\frac{1}{8}x^{6}-3x^{3}+24-64x^{-3}$$