Question

simplifying a division expression
choose two correct answers that are equivalent to \\(\frac{a^2}{a^9}\\). assume \\(a \
eq 0\\).
options: \\(\frac{1}{a^{-7}}\\), \\(-a^{-7}\\), \\(-a^7\\), \\(\frac{1}{a^7}\\), \\(a^{-7}\\)

Explanation:

Step1: Simplify the original expression

To simplify \(\frac{a^2}{a^9}\), we use the quotient rule for exponents, which states that \(\frac{a^m}{a^n}=a^{m - n}\) (where \(a
eq0\)). Here, \(m = 2\) and \(n=9\), so \(\frac{a^2}{a^9}=a^{2 - 9}=a^{-7}\).

Step2: Recall the definition of negative exponents

The definition of a negative exponent is \(a^{-n}=\frac{1}{a^n}\) (where \(a
eq0\) and \(n\) is a positive integer). So, \(a^{-7}=\frac{1}{a^7}\).

Now we check the options:

• Option \(\frac{1}{a^{-7}}\): Using the negative exponent rule, \(\frac{1}{a^{-7}}=a^{7}\), which is not equivalent.
• Option \(-a^{-7}\): This is \(-\times a^{-7}\), which is not equivalent to \(a^{-7}\) or \(\frac{1}{a^7}\) (unless \(a = - 1\), but generally no).
• Option \(-a^{7}\): This is \(-a^{7}\), not equivalent.
• Option \(\frac{1}{a^{7}}\): From the negative exponent rule, \(a^{-7}=\frac{1}{a^7}\), so this is equivalent.
• Option \(a^{-7}\): This is the result from Step 1, so this is equivalent.

Answer:

The two correct answers are \(a^{-7}\) and \(\frac{1}{a^{7}}\) (corresponding to the bottom - middle and bottom - left options, or as per the visual labels, the option with \(a^{-7}\) and the option with \(\frac{1}{a^{7}}\)).