For the function $g(t)=10\left(\frac{1}{5}\right)^{t+1}$, what is $g(-3)$?
The given function is \(g(t) = 10\left(\frac{1}{5}\right)^{t+1}\). We are tasked with evaluating this function at \(t = -3\). This involves substituting the value of \(t\) into the expression and performing the necessary arithmetic operations according to the rules of exponents and order of operations.
First, we substitute \(t = -3\) into the expression for \(g(t)\):
\[g(-3) = 10\left(\frac{1}{5}\right)^{-3+1}\]
Next, we simplify the exponent:
\[-3 + 1 = -2\]
So, the expression becomes:
\[g(-3) = 10\left(\frac{1}{5}\right)^{-2}\]
To evaluate \(\left(\frac{1}{5}\right)^{-2}\), we use the property of exponents that \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\).
Applying this:
\[\left(\frac{1}{5}\right)^{-2} = \left(\frac{5}{1}\right)^{2} = 5^2\]
Calculating the square:
\[5^2 = 25\]
Now, substitute this back into the expression for \(g(-3)\):
\[g(-3) = 10 \times 25\]
Multiplying these values yields:
\[g(-3) = 250\]
The final result is \(\boxed{250}\).
Exponential functions like \(g(t)\) are used in real-life to model various processes involving rapid change. For instance, consider a chemical substance that decays over time, where the concentration is represented by a function of time \(t\). If the amount remaining is given by a base smaller than 1 raised to a power, it indicates a decrease. Understanding how to evaluate these functions at specific points in time helps scientists and researchers predict the amount of substance remaining in a sample after a certain interval, ensuring accuracy in experiments and environmental studies.