Powers: Exponential Equations
Correct.
[[Snippet]] First, convert the negative power of $$3$$ into a positive one (i.e., $$3^{-5x} = \frac{1}{3^{5x}}$$). >$$\displaystyle \frac{1}{3^{5x}} < \frac{1}{243^2}$$ Cross multiply to get $$243^2 < 3^{5x}$$. Express everything as a power of $$3$$. >$$243 = 9 \cdot 27 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 3^5$$ So $$243^2 < 3^{5x}$$ can be expressed as >$$(3^{5})^{2} < 3^{5x}$$ >$$3^{10} < 3^{5x}$$. Since the bases are the same, the powers can be compared directly. >$$10 < 5x$$ >$$2 < x$$ Hence, the smallest possible integer value of $$x$$ is $$3$$.Incorrect.
[[Snippet]] It's true that $$x=6$$ will indeed make the left side smaller than the right side, but did you choose this answer because you missed out on the fact that the question asks for the smallest possible value of $$x$$?Incorrect.
[[Snippet]]Incorrect.
[[Snippet]]Incorrect.
Note that the question uses an inequality, not an equation. If you've done your calculation right, $$x$$ cannot equal $$2$$. [[Snippet]]Care to leave a comment and tell us about it?