Solve frac{(a^{2})^{6}*(a^{3})^{8}}{(a^2)^3}=(a^6)^5

$$\frac { ( a ^ { 2 } ) ^ { 6 } \times ( a ^ { 3 } ) ^ { 8 } } { ( a ^ { 2 } ) ^ { 3 } } = ( a ^ { 6 } ) ^ { 5 }$$

Solve for a

$$a\neq 0$$

Solution Steps

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a^{6}.

$$\left(a^{2}\right)^{6}\left(a^{3}\right)^{8}=a^{6}\left(a^{6}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$a^{12}\left(a^{3}\right)^{8}=a^{6}\left(a^{6}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 8 to get 24.

$$a^{12}a^{24}=a^{6}\left(a^{6}\right)^{5}$$

To multiply powers of the same base, add their exponents. Add 12 and 24 to get 36.

$$a^{36}=a^{6}\left(a^{6}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 5 to get 30.

$$a^{36}=a^{6}a^{30}$$

To multiply powers of the same base, add their exponents. Add 6 and 30 to get 36.

$$a^{36}=a^{36}$$

Subtract a^{36} from both sides.

$$a^{36}-a^{36}=0$$

Combine a^{36} and -a^{36} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Variable a cannot be equal to 0.

$$a\in \mathrm{R}\setminus 0$$