solve: $left(\frac{1}{8}\right)^{-3a}=512^{3a}$\n$a = - 8$\n$a = 0$\n$a = 8$\nno solution

Explanation:

Step1: Rewrite bases as powers of 2

Since $\frac{1}{8}=2^{-3}$ and $512 = 2^{9}$, the equation $\left(\frac{1}{8}\right)^{-3a}=512^{3a}$ becomes $(2^{-3})^{-3a}=(2^{9})^{3a}$.

Step2: Apply power - of - a - power rule

The power - of - a - power rule $(x^{m})^{n}=x^{mn}$. So, $(2^{-3})^{-3a}=2^{(-3)\times(-3a)} = 2^{9a}$ and $(2^{9})^{3a}=2^{9\times3a}=2^{27a}$. The equation is now $2^{9a}=2^{27a}$.

Step3: Set exponents equal

If $x^{m}=x^{n}$, then $m = n$ for $x>0,x\neq1$. So, $9a = 27a$.

Step4: Solve for a

Subtract $9a$ from both sides: $0=27a - 9a$, which simplifies to $0 = 18a$. Dividing both sides by 18 gives $a = 0$.

Answer:

$a = 0$