To simplify the expression ( \frac{3^{18} \cdot 7^{18}}{21^{19}} ), we first rewrite ( 21 ) in terms of its prime factors:
[
21 = 3 \cdot 7
]
Now, we can express ( 21^{19} ) as:
[
21^{19} = (3 \cdot 7)^{19} = 3^{19} \cdot 7^{19}
]
Next, we can substitute this back into the original expression:
[
\frac{3^{18} \cdot 7^{18}}{21^{19}} = \frac{3^{18} \cdot 7^{18}}{3^{19} \cdot 7^{19}}
]
Now, we can simplify the fraction by applying the rules of exponents. For ( 3^{18} ) over ( 3^{19} ), we have:
[
\frac{3^{18}}{3^{19}} = 3^{18 - 19} = 3^{-1}
]
For ( 7^{18} ) over ( 7^{19} ), we have:
[
\frac{7^{18}}{7^{19}} = 7^{18 - 19} = 7^{-1}
]
Combining these results together, we get:
[
\frac{3^{18} \cdot 7^{18}}{3^{19} \cdot 7^{19}} = 3^{-1} \cdot 7^{-1} = \frac{1}{3} \cdot \frac{1}{7} = \frac{1}{21}
]
Thus, the final answer is:
[
\frac{3^{18} \cdot 7^{18}}{21^{19}} = \frac{1}{21}
]
