solve: $$ ( \frac { 1 } { 8 } ) ^ { - 3 a } = 5 1 2 ^ { 3 a } $$\n$$ a = - 8 $$\n$$ a = 0 $$\n$$ a = 8…

solve: $$ ( \frac { 1 } { 8 } ) ^ { - 3 a } = 5 1 2 ^ { 3 a } $$\n$$ a = - 8 $$\n$$ a = 0 $$\n$$ a = 8 $$\nno solution

solve: $$ ( \frac { 1 } { 8 } ) ^ { - 3 a } = 5 1 2 ^ { 3 a } $$\n$$ a = - 8 $$\n$$ a = 0 $$\n$$ a = 8 $$\nno solution

Answer

Explanation:

Step1: Rewrite the bases

Rewrite (\frac{1}{8}) as (2^{-3}) and (512) as (2^{9}). The equation (\left(\frac{1}{8}\right)^{-3a}=512^{3a}) becomes ((2^{-3})^{-3a}=(2^{9})^{3a}).

Step2: Apply the power - of - a - power rule

Using ((a^{m})^{n}=a^{mn}), we have (2^{(-3)\times(-3a)} = 2^{9\times3a}), which simplifies to (2^{9a}=2^{27a}).

Step3: Set the exponents equal

Since the bases are the same ((a^{m}=a^{n}) implies (m = n) for (a>0,a\neq1)), we set (9a = 27a). Subtract (9a) from both sides: (9a-9a=27a - 9a), getting (0 = 18a). Divide both sides by (18): (a=\frac{0}{18}=0).

Answer:

(a = 0)