2100.35 – Fiddling with Exponents

Express $8^x$ in terms of $a$ , given that $a = 2^{(x+2)}$ .

Solution

$a = 2^{(x+2)}\rightarrow a^3=2^{(x+2)\cdot3}=2^{(3x+6)}$

So $\frac{a^3}{2^6} = \frac{2^{(3x+6)}}{2^6}=2^{(3x+6-6)} =2^{3x} = (2^3)^x = 8^x.\\ \therefore 8^x = \frac{a^3}{64}.$