Solve frac{(x^{-2})^{2^{2}}*[(-x^{-2})^{3^{2^{2}}}]^{-1}}{x^{2^{3}}*[(-x^{3})^3^2]^2^3}

$$\frac { ( x ^ { - 2 } ) ^ { 2 ^ { 2 } } \cdot [ ( - x ^ { - 2 } ) ^ { 3 ^ { 2 ^ { 2 } } } ] ^ { - 1 } } { x ^ { 2 ^ { 3 } } \cdot [ ( - x ^ { 3 } ) ^ { 3 ^ { 2 } } ] ^ { 2 ^ { 3 } } }$$

Evaluate

$$-\frac{1}{x^{70}}$$

Short Solution Steps

Calculate 2 to the power of 2 and get 4.

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

Calculate 2 to the power of 2 and get 4.

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

Calculate 3 to the power of 4 and get 81.

$$\frac{\left(x^{-2}\right)^{4}\left(\left(-x^{-2}\right)^{81}\right)^{-1}}{x^{2^{3}}\left(\left(-x^{3}\right)^{3^{2}}\right)^{2^{3}}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\left(x^{-2}\right)^{4}\left(\left(-x^{-2}\right)^{81}\right)^{-1}}{x^{8}\left(\left(-x^{3}\right)^{3^{2}}\right)^{2^{3}}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(x^{-2}\right)^{4}\left(\left(-x^{-2}\right)^{81}\right)^{-1}}{x^{8}\left(\left(-x^{3}\right)^{9}\right)^{2^{3}}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\left(x^{-2}\right)^{4}\left(\left(-x^{-2}\right)^{81}\right)^{-1}}{x^{8}\left(\left(-x^{3}\right)^{9}\right)^{8}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 4 to get -8.

$$\frac{x^{-8}\left(\left(-x^{-2}\right)^{81}\right)^{-1}}{x^{8}\left(\left(-x^{3}\right)^{9}\right)^{8}}$$

To raise a power to another power, multiply the exponents. Multiply 81 and -1 to get -81.

$$\frac{x^{-8}\left(-x^{-2}\right)^{-81}}{x^{8}\left(\left(-x^{3}\right)^{9}\right)^{8}}$$

To raise a power to another power, multiply the exponents. Multiply 9 and 8 to get 72.

$$\frac{x^{-8}\left(-x^{-2}\right)^{-81}}{x^{8}\left(-x^{3}\right)^{72}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\left(-x^{-2}\right)^{-81}}{x^{16}\left(-x^{3}\right)^{72}}$$

Expand \left(-x^{-2}\right)^{-81}.

$$\frac{\left(-1\right)^{-81}\left(x^{-2}\right)^{-81}}{x^{16}\left(-x^{3}\right)^{72}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -81 to get 162.

$$\frac{\left(-1\right)^{-81}x^{162}}{x^{16}\left(-x^{3}\right)^{72}}$$

Calculate -1 to the power of -81 and get -1.

$$\frac{-x^{162}}{x^{16}\left(-x^{3}\right)^{72}}$$

Expand \left(-x^{3}\right)^{72}.

$$\frac{-x^{162}}{x^{16}\left(-1\right)^{72}\left(x^{3}\right)^{72}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 72 to get 216.

$$\frac{-x^{162}}{x^{16}\left(-1\right)^{72}x^{216}}$$

Calculate -1 to the power of 72 and get 1.

$$\frac{-x^{162}}{x^{16}\times 1x^{216}}$$

To multiply powers of the same base, add their exponents. Add 16 and 216 to get 232.

$$\frac{-x^{162}}{x^{232}\times 1}$$

Cancel out x^{162} in both numerator and denominator.

$$\frac{-1}{x^{70}}$$

Differentiate w.r.t. x

$$\frac{70}{x^{71}}$$