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)`