Simplify the following expression:
$\frac{\frac{(-3)^5}{5^3 \cdot 5^9}}{5}$
We are given the expression \(\frac{\frac{(-3)^5}{5^3 \cdot 5^9}}{5}\). To simplify this, we first focus on the denominator. Using the rule for multiplication of powers with the same base, \(a^m \cdot a^n = a^{m+n}\), we can simplify the denominator of the fraction \(\frac{(-3)^5}{5^3 \cdot 5^9}\) by noting that \(5^3 \cdot 5^9 = 5^{3+9} = 5^{12}\). The entire expression can be rewritten as \(\frac{(-3)^5}{5^{12}} \cdot \frac{1}{5}\).
Now, we combine the terms in the denominator. Applying the exponent rule again, we have \(5^{12} \cdot 5 = 5^{12} \cdot 5^1 = 5^{12+1} = 5^{13}\). The expression now becomes \(\frac{(-3)^5}{5^{13}}\). Since \(-3\) raised to an odd power remains negative, \((-3)^5 = - (3^5) = -243\). Thus, the expression is \(-\frac{3^5}{5^{13}}\).
In physics, especially when dealing with radioactive decay or certain scaling laws, we often encounter complex fractions with various powers of a base. Simplifying these expressions is crucial to understanding the magnitude of the results. For example, if a model predicts that the intensity of a signal scales as \(\frac{I_0}{D}\) where \(D\) is a complex product of distance factors, performing this type of simplification allows for easier calculation of the final intensity, helping engineers design more efficient communication systems.