Solve 9^{8}*frac{(x^{2})^5}{(27)^4}*(x^3)^2

$$9 ^ { 8 } \times \frac { ( x ^ { 2 } ) ^ { 5 } } { ( 27 ) ^ { 4 } } \times ( x ^ { 3 } ) ^ { 2 }$$

Evaluate

$$81x^{16}$$

Solution Steps

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

$$9^{8}\times \frac{x^{10}}{27^{4}}\left(x^{3}\right)^{2}$$

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

$$9^{8}\times \frac{x^{10}}{27^{4}}x^{6}$$

Calculate 9 to the power of 8 and get 43046721.

$$43046721\times \frac{x^{10}}{27^{4}}x^{6}$$

Calculate 27 to the power of 4 and get 531441.

$$43046721\times \frac{x^{10}}{531441}x^{6}$$

Cancel out 531441, the greatest common factor in 43046721 and 531441.

$$81x^{10}x^{6}$$

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

$$81x^{16}$$

Differentiate w.r.t. x

$$1296x^{15}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(9^{8}\times \frac{x^{10}}{27^{4}}\left(x^{3}\right)^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(9^{8}\times \frac{x^{10}}{27^{4}}x^{6})$$

Calculate 9 to the power of 8 and get 43046721.

$$\frac{\mathrm{d}}{\mathrm{d}x}(43046721\times \frac{x^{10}}{27^{4}}x^{6})$$

Calculate 27 to the power of 4 and get 531441.

$$\frac{\mathrm{d}}{\mathrm{d}x}(43046721\times \frac{x^{10}}{531441}x^{6})$$

Cancel out 531441, the greatest common factor in 43046721 and 531441.

$$\frac{\mathrm{d}}{\mathrm{d}x}(81x^{10}x^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(81x^{16})$$

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

$$16\times 81x^{16-1}$$

Multiply 16 times 81.

$$1296x^{16-1}$$

Subtract 1 from 16.

$$1296x^{15}$$