Question

36^x+1/2•8^x+1 ÷ 6^x-1 •12^x+5•4^x-1

Answer 1

1/216

Explanation:

You want to simplify the rational expression 36^(x+1/2)•8^(x+1) ÷ (6^(x-1) •12^(x+5)•4^(x-1)).

Rules of exponents

The relevant rules of exponents are ...

(a^b)(c^b) = (ac)^b

a^(b+c) = (a^b)(a^c)

Application

`(36^(x+1/2)8^(x+1))/(6^(x-1)12^(x+5)4^(x-1))=\left((36^(1/2)\cdot8^1)/(6^(-1)\cdot12^5\cdot 4^(-1))\right)\left((36\cdot8)/(6\cdot12\cdot4)\right)^x\\\\=(48\cdot24)/(12^5)\cdot\left((288)/(288)\right)^x=\boxed{(1)/(216)}`