Question

Which expressions are equivalent to the one below? Check all that apply. 8^x
a) (32/4)^x b) x^4 c) 8*8^x-1 d) 32^x/4 e) 32^x/4^x f)8*8^x+1

Answer 1

$\bf a)\qquad \left( \cfrac{32}{4} \right)^x\implies \left( \cfrac{8}{1} \right)^x\implies (8)^x\implies 8^x \\\\\\ b)\qquad x^4\impliedby \textit{nothing to simplify} \\\\\\ c)\qquad 8\cdot 8^{x-1}\implies 8^1\cdot 8^{x-1}\impliedby \textit{same base, add the exponents} \\\\\\ \left. \qquad \right.\left. \qquad \right.8^{1+x-1}\implies 8^x$

$\bf d)\qquad \cfrac{32^x}{4}\implies \cfrac{(2^5)^x}{2^2}\implies (2^5)^x\cdot 2^{-2}\implies 2^{5x}\cdot 2^{-2}\implies 2^{5x-2} \\\\\\ e)\qquad \cfrac{32^x}{4^x}\implies \left( \cfrac{32}{4} \right)^x\implies \left( \cfrac{8}{1} \right)^x\implies 8^x \\\\\\ f)\qquad 8\cdot 8^{x+1}\implies 8^1\cdot 8^{x+1}\implies 8^{1+x+1}\implies 8^{x+2}$