Question

evaluate. write your answer as a whole number or as a simplified fraction.\\(\frac{3^{-9}}{3^{-6}} = \frac{quad}{quad}\\)

Explanation:

Step1: Apply exponent quotient rule

When dividing exponents with the same base, subtract the denominator exponent from the numerator exponent: $a^m / a^n = a^{m-n}$
$\frac{3^{-9}}{3^{-6}} = 3^{-9 - (-6)}$

Step2: Simplify the exponent

Calculate the value of the exponent:
$3^{-9 + 6} = 3^{-3}$

Step3: Rewrite negative exponent as fraction

A negative exponent means the reciprocal of the positive exponent: $a^{-k} = \frac{1}{a^k}$
$3^{-3} = \frac{1}{3^3}$

Step4: Compute the denominator

Calculate the value of $3^3$:
$\frac{1}{3 \times 3 \times 3} = \frac{1}{27}$

Answer:

$\frac{1}{27}$