1. **State the problem:** We want to find the predicted value of a car in 2016, given it depreciates exponentially from $23,000 in 2008 to $6,400 in 2013. 2. **Formula for exponential decay:** The value $V$ at time $t$ is given by $$V = V_0 \times r^{t}$$ where $V_0$ is the initial value, $r$ is the decay rate per year, and $t$ is the number of years since the initial time. 3. **Identify known values:** - Initial year: 2008, so $t=0$ at 2008 - $V_0 = 23000$ - In 2013, $t=2013-2008=5$, value $V=6400$ 4. **Find decay rate $r$:** $$6400 = 23000 \times r^{5}$$ Divide both sides by 23000: $$\frac{6400}{23000} = r^{5}$$ $$\cancel{\frac{6400}{23000}} = r^{5}$$ Simplify fraction: $$\frac{64}{230} = r^{5}$$ 5. **Solve for $r$:** Take the fifth root: $$r = \left(\frac{64}{230}\right)^{\frac{1}{5}}$$ Calculate approximate value: $$r \approx 0.749$$ 6. **Find value in 2016:** $t=2016-2008=8$ $$V = 23000 \times 0.749^{8}$$ Calculate: $$V \approx 23000 \times 0.100 = 2300$$ 7. **Final answer:** The predicted value of the car in 2016 is approximately **2300** dollars.