Solve [(a^{3})^{3}*(a^{4})^{3}]div(a^3)^2div(a^2)^3

$$[ ( a ^ { 3 } ) ^ { 3 } \cdot ( a ^ { 4 } ) ^ { 3 } ] \div ( a ^ { 3 } ) ^ { 2 } \div ( a ^ { 2 } ) ^ { 3 }$$

Evaluate

$$a^{9}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{\frac{a^{9}\left(a^{4}\right)^{3}}{\left(a^{3}\right)^{2}}}{\left(a^{2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{\frac{a^{9}a^{12}}{\left(a^{3}\right)^{2}}}{\left(a^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 9 and 12 to get 21.

$$\frac{\frac{a^{21}}{\left(a^{3}\right)^{2}}}{\left(a^{2}\right)^{3}}$$

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

$$\frac{\frac{a^{21}}{a^{6}}}{\left(a^{2}\right)^{3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 6 from 21 to get 15.

$$\frac{a^{15}}{\left(a^{2}\right)^{3}}$$

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

$$\frac{a^{15}}{a^{6}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 6 from 15 to get 9.

$$a^{9}$$

Differentiate w.r.t. a

$$9a^{8}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{9}\left(a^{4}\right)^{3}}{\left(a^{3}\right)^{2}}}{\left(a^{2}\right)^{3}})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{9}a^{12}}{\left(a^{3}\right)^{2}}}{\left(a^{2}\right)^{3}})$$

To multiply powers of the same base, add their exponents. Add 9 and 12 to get 21.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{21}}{\left(a^{3}\right)^{2}}}{\left(a^{2}\right)^{3}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{21}}{a^{6}}}{\left(a^{2}\right)^{3}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 6 from 21 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{15}}{\left(a^{2}\right)^{3}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{15}}{a^{6}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 6 from 15 to get 9.

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

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

$$9a^{9-1}$$

Subtract 1 from 9.

$$9a^{8}$$