Question

which statement is true?
a (\frac{3^{-15}}{3^{7}} < left(3^{-8}
ight)cdotleft(3^{-9}
ight))
b (\frac{3^{-15}}{3^{7}} > left(3^{-8}
ight)cdotleft(3^{-9}
ight))
c (\frac{3^{-15}}{3^{7}} = left(3^{-8}
ight)cdotleft(3^{-9}
ight))

Explanation:

Step1: Simplify left-hand side

When dividing exponents with the same base, subtract exponents:
$\frac{3^{-15}}{3^{7}} = 3^{-15-7} = 3^{-22}$

Step2: Simplify right-hand side

When multiplying exponents with the same base, add exponents:
$(3^{-8})\cdot(3^{-9}) = 3^{-8+(-9)} = 3^{-17}$

Step3: Compare the two values

For a base $b>1$, smaller exponents give smaller values. Since $-22 < -17$, we have $3^{-22} > 3^{-17}$, so $\frac{3^{-15}}{3^{7}} > (3^{-8})\cdot(3^{-9})$.

Answer:

B. $\frac{3^{-15}}{3^{7}} > (3^{-8})\cdot(3^{-9})$