Question

which expression is equivalent to $(5^{-9})^{-9}$?
$\frac{1}{5^{-18}}$ $5^{3} \cdot 5^{-6}$ $\frac{1}{5^{-81}}$ $5^{-18}$

Explanation:

Step1: Apply exponent power rule

When raising a power to a power, multiply the exponents: $(a^m)^n = a^{m \times n}$.
For $(5^{-9})^{-9}$, calculate $(-9) \times (-9) = 81$, so $(5^{-9})^{-9} = 5^{81}$.

Step2: Simplify each option

• Option1: $\frac{1}{5^{-18}} = 5^{18}$ (since $\frac{1}{a^{-k}}=a^k$)
• Option2: $5^3 \cdot 5^{-6} = 5^{3+(-6)} = 5^{-3}$ (product rule: $a^m \cdot a^n=a^{m+n}$)
• Option3: $\frac{1}{5^{-81}} = 5^{81}$ (since $\frac{1}{a^{-k}}=a^k$)
• Option4: $5^{-18}$ (already simplified, does not match $5^{81}$)

Answer:

$\frac{1}{5^{-81}}$