Question

select all that are equivalent to $\frac{a^{2}}{a^{9}}$. assume $a \
eq 0$.
$a^{-7}$
$-a^{-7}$
$\frac{1}{a^{7}}$
$-a^{7}$
$\frac{1}{a^{-7}}$

Explanation:

Step1: Apply exponent subtraction rule

When dividing like bases, subtract exponents: $\frac{a^m}{a^n}=a^{m-n}$.
For $\frac{a^2}{a^9}$, this gives $a^{2-9}=a^{-7}$.

Step2: Rewrite negative exponent as positive

A negative exponent means reciprocal: $a^{-n}=\frac{1}{a^n}$.
So $a^{-7}=\frac{1}{a^7}$.

Step3: Evaluate other options

• $-a^{-7}$ introduces an unnecessary negative sign, not equivalent.
• $-a^7$ is a negative positive exponent term, not equivalent.
• $\frac{1}{a^{-7}}=a^7$, which is the inverse of the original result, not equivalent.

Answer:

$a^{-7}$, $\frac{1}{a^7}$