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

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

Evaluate

$$81x^{4}$$

Solution Steps

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

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

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

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

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

$$\frac{9^{8}x^{4}}{27^{4}}$$

Calculate 9 to the power of 8 and get 43046721.

$$\frac{43046721x^{4}}{27^{4}}$$

Calculate 27 to the power of 4 and get 531441.

$$\frac{43046721x^{4}}{531441}$$

Divide 43046721x^{4} by 531441 to get 81x^{4}.

$$81x^{4}$$

Differentiate w.r.t. x

$$324x^{3}$$

Steps Using Definition of a Derivative

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

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

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

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

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

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

Calculate 9 to the power of 8 and get 43046721.

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

Calculate 27 to the power of 4 and get 531441.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{43046721x^{4}}{531441})$$

Divide 43046721x^{4} by 531441 to get 81x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(81x^{4})$$

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

$$4\times 81x^{4-1}$$

Multiply 4 times 81.

$$324x^{4-1}$$

Subtract 1 from 4.

$$324x^{3}$$