Solve frac{(x^{2})^{3}*x^{2^{4}}*x^{-2^{3}}}{x^{-4^2}*x^(-3)^2*x^12}

$$\frac { ( x ^ { 2 } ) ^ { 3 } \cdot x ^ { 2 ^ { 4 } } \cdot x ^ { - 2 ^ { 3 } } } { x ^ { - 4 ^ { 2 } } \cdot x ^ { ( - 3 ) ^ { 2 } } \cdot x ^ { 12 } }$$

Evaluate

$$x^{9}$$

Solution Steps

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

$$\frac{x^{6}x^{2^{4}}x^{-2^{3}}}{x^{-4^{2}}x^{\left(-3\right)^{2}}x^{12}}$$

Cancel out x^{6} in both numerator and denominator.

$$\frac{x^{-2^{3}}x^{2^{4}}}{x^{-4^{2}}x^{6}x^{\left(-3\right)^{2}}}$$

Calculate 2 to the power of 3 and get 8.

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

Calculate 2 to the power of 4 and get 16.

$$\frac{x^{-8}x^{16}}{x^{-4^{2}}x^{6}x^{\left(-3\right)^{2}}}$$

To multiply powers of the same base, add their exponents. Add -8 and 16 to get 8.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{x^{8}}{x^{-16}x^{6}x^{\left(-3\right)^{2}}}$$

To multiply powers of the same base, add their exponents. Add -16 and 6 to get -10.

$$\frac{x^{8}}{x^{-10}x^{\left(-3\right)^{2}}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{x^{8}}{x^{-10}x^{9}}$$

To multiply powers of the same base, add their exponents. Add -10 and 9 to get -1.

$$\frac{x^{8}}{x^{-1}}$$

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

$$x^{9}$$

Differentiate w.r.t. x

$$9x^{8}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{6}x^{2^{4}}x^{-2^{3}}}{x^{-4^{2}}x^{\left(-3\right)^{2}}x^{12}})$$

Cancel out x^{6} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-2^{3}}x^{2^{4}}}{x^{-4^{2}}x^{6}x^{\left(-3\right)^{2}}})$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-8}x^{2^{4}}}{x^{-4^{2}}x^{6}x^{\left(-3\right)^{2}}})$$

Calculate 2 to the power of 4 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-8}x^{16}}{x^{-4^{2}}x^{6}x^{\left(-3\right)^{2}}})$$

To multiply powers of the same base, add their exponents. Add -8 and 16 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{x^{-4^{2}}x^{6}x^{\left(-3\right)^{2}}})$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{x^{-16}x^{6}x^{\left(-3\right)^{2}}})$$

To multiply powers of the same base, add their exponents. Add -16 and 6 to get -10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{x^{-10}x^{\left(-3\right)^{2}}})$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{x^{-10}x^{9}})$$

To multiply powers of the same base, add their exponents. Add -10 and 9 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{8}}{x^{-1}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{9})$$

The derivative of ax^{n} is nax^{n-1}.

$$9x^{9-1}$$

Subtract 1 from 9.

$$9x^{8}$$