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Which option is equivalent to StartFraction left-parenthesis 9 right-parenthesis Superscript negative 4 Baseline over left-parenthesis negative 3 right-parenthesis Superscript negative 8 Baseline EndFraction?

Answer options with 4 options
A.
1

B.
negative 1

C.
StartFraction 1 over 3 Superscript 4 Baseline EndFraction

D.
negative StartFraction 1 over 3 Superscript 4 Baseline EndFraction

1 answer

To simplify the expression \(\frac{9^{-4}}{(-3)^{-8}}\), we can use the properties of exponents.

  1. Recall that \(a^{-n} = \frac{1}{a^n}\), so we can rewrite the expression: \[ \frac{9^{-4}}{(-3)^{-8}} = \frac{1}{9^4} \div \frac{1}{(-3)^8} = \frac{(-3)^8}{9^4} \]

  2. Now we can calculate the powers:

    • \(9 = 3^2\), therefore: \[ 9^4 = (3^2)^4 = 3^{8} \]
    • \((-3)^8 = 3^8\) (since a negative number raised to an even power is positive).

  3. Substitute back: \[ \frac{(-3)^8}{9^4} = \frac{3^8}{3^8} = 1 \]

Thus, the equivalent expression is: \[ \boxed{1} \]

So the correct answer is A. 1.