Question

which expression is equivalent to \\(\frac{3^{-10}}{3^{5}}\\)?\
\\((3^{-7})^{5}\\) \\(3^{15}\\) \\(3^{-8}\cdot 3^{-7}\\) \\(3^{-5}\\)

Explanation:

Step1: Apply quotient rule for exponents

When dividing like bases, subtract exponents: $\frac{a^m}{a^n}=a^{m-n}$.
$\frac{3^{-10}}{3^5}=3^{-10-5}=3^{-15}$

Step2: Evaluate each option

Option1: Simplify $(3^{-7})^5$

Use power of a power rule: $(a^m)^n=a^{m \cdot n}$.
$(3^{-7})^5=3^{-7 \times 5}=3^{-35}$

Option2: $3^{15}$

This is $3^{15}$, not $3^{-15}$.

Option3: Simplify $3^{-8} \cdot 3^{-7}$

Use product rule for exponents: $a^m \cdot a^n=a^{m+n}$.
$3^{-8} \cdot 3^{-7}=3^{-8+(-7)}=3^{-15}$

Option4: $3^{-5}$

This is $3^{-5}$, not $3^{-15}$.

Answer:

C. $3^{-8} \cdot 3^{-7}$