which statement is true? \na (\frac{3^{-15}}{3^{7}} < left(3^{-8}\right)cdotleft(3^{-9}\right)) \nb…

Explanation:

Step1: Simplify left - hand side

For the expression (\frac{3^{- 15}}{3^{7}}), according to the rule of exponents (a^{m}\div a^{n}=a^{m - n}), we have: (\frac{3^{-15}}{3^{7}}=3^{-15 - 7}=3^{-22})

Step2: Simplify right - hand side

For the expression ((3^{-8})\cdot(3^{-9})), according to the rule of exponents (a^{m}\cdot a^{n}=a^{m + n}), we have: ((3^{-8})\cdot(3^{-9})=3^{-8+( - 9)}=3^{-17})

Step3: Compare the two results

We know that for the exponential function (y = 3^{x}), when the base (a=3>1), the function is an increasing function. That is, when (x_1<x_2), (3^{x_1}<3^{x_2}). For negative exponents, (3^{-22}=\frac{1}{3^{22}}) and (3^{-17}=\frac{1}{3^{17}}). Since (3^{22}>3^{17}), then (\frac{1}{3^{22}}<\frac{1}{3^{17}}), so (3^{-22}<3^{-17}), which means (\frac{3^{-15}}{3^{7}}<(3^{-8})\cdot(3^{-9}))

Answer:

A. (\boldsymbol{\frac{3^{-15}}{3^{7}}<\left(3^{-8}\right)\cdot\left(3^{-9}\right)})