Question

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

Explanation:

Step1: Apply exponent rule for division

When dividing exponents with the same base, we subtract the exponents: \( \frac{a^m}{a^n} = a^{m - n} \). Here, \( a = 6 \), \( m = 6 \), and \( n = 9 \). So we have \( 6^{6 - 9} \).

Step2: Calculate the exponent

Calculate \( 6 - 9 = -3 \), so we get \( 6^{-3} \).

Step3: Rewrite negative exponent as positive reciprocal

A negative exponent means the reciprocal with the positive exponent: \( a^{-n} = \frac{1}{a^n} \). So \( 6^{-3} = \frac{1}{6^3} \).

Step4: Calculate \( 6^3 \)

\( 6^3 = 6\times6\times6 = 216 \), so \( \frac{1}{6^3} = \frac{1}{216} \).

Answer:

\(\frac{1}{216}\)