Solve frac{(a^{6})^{14}*a^{8}*(a^{9})^{16}}{(a^{10})^8*(a^9)^11*a^53}*8

$$\frac { ( a ^ { 6 } ) ^ { 14 } \cdot a ^ { 8 } \cdot ( a ^ { 9 } ) ^ { 16 } } { ( a ^ { 10 } ) ^ { 8 } \cdot ( a ^ { 9 } ) ^ { 11 } \cdot a ^ { 53 } } \cdot 8$$

Evaluate

$$8a^{4}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 6 and 14 to get 84.

$$\frac{a^{84}a^{8}\left(a^{9}\right)^{16}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8$$

To raise a power to another power, multiply the exponents. Multiply 9 and 16 to get 144.

$$\frac{a^{84}a^{8}a^{144}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8$$

To multiply powers of the same base, add their exponents. Add 84 and 8 to get 92.

$$\frac{a^{92}a^{144}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8$$

To multiply powers of the same base, add their exponents. Add 92 and 144 to get 236.

$$\frac{a^{236}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8$$

To raise a power to another power, multiply the exponents. Multiply 10 and 8 to get 80.

$$\frac{a^{236}}{a^{80}\left(a^{9}\right)^{11}a^{53}}\times 8$$

To raise a power to another power, multiply the exponents. Multiply 9 and 11 to get 99.

$$\frac{a^{236}}{a^{80}a^{99}a^{53}}\times 8$$

To multiply powers of the same base, add their exponents. Add 80 and 99 to get 179.

$$\frac{a^{236}}{a^{179}a^{53}}\times 8$$

To multiply powers of the same base, add their exponents. Add 179 and 53 to get 232.

$$\frac{a^{236}}{a^{232}}\times 8$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 232 from 236 to get 4.

$$a^{4}\times 8$$

Differentiate w.r.t. a

$$32a^{3}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 6 and 14 to get 84.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{84}a^{8}\left(a^{9}\right)^{16}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8)$$

To raise a power to another power, multiply the exponents. Multiply 9 and 16 to get 144.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{84}a^{8}a^{144}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8)$$

To multiply powers of the same base, add their exponents. Add 84 and 8 to get 92.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{92}a^{144}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8)$$

To multiply powers of the same base, add their exponents. Add 92 and 144 to get 236.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{236}}{\left(a^{10}\right)^{8}\left(a^{9}\right)^{11}a^{53}}\times 8)$$

To raise a power to another power, multiply the exponents. Multiply 10 and 8 to get 80.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{236}}{a^{80}\left(a^{9}\right)^{11}a^{53}}\times 8)$$

To raise a power to another power, multiply the exponents. Multiply 9 and 11 to get 99.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{236}}{a^{80}a^{99}a^{53}}\times 8)$$

To multiply powers of the same base, add their exponents. Add 80 and 99 to get 179.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{236}}{a^{179}a^{53}}\times 8)$$

To multiply powers of the same base, add their exponents. Add 179 and 53 to get 232.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{236}}{a^{232}}\times 8)$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 232 from 236 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{4}\times 8)$$

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

$$4\times 8a^{4-1}$$

Multiply 4 times 8.

$$32a^{4-1}$$

Subtract 1 from 4.

$$32a^{3}$$