A new car is purchased for 16400 dollars. The value of the car depreciates at $$ 8.25 \% $$ per year. What will the value of the car be, to the nearest cent, after 10 years?

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To find the value of the car after 10 years, we can use the formula for depreciation:

$$ V = P \times (1 - r)^n $$

where:
- $$ V $$ is the value of the car after $$ n $$ years,
- $$ P $$ is the initial value of the car,
- $$ r $$ is the annual depreciation rate,
- $$ n $$ is the number of years.

Given:
- Initial value of the car, $$ P = 16400 $$ dollars,
- Annual depreciation rate, $$ r = 8.25\% = 0.0825 $$,
- Number of years, $$ n = 10 $$.

Substitute the values into the formula:

$$ V = 16400 \times (1 - 0.0825)^{10} $$

Now, we can calculate the value of the car after 10 years.
Calculate the value by following steps:
- step0: Calculate:
  $$16400\left(1-0.0825\right)^{10}$$
- step1: Subtract the numbers:
  $$16400\times 0.9175^{10}$$
- step2: Convert the expressions:
  $$16400\left(\frac{367}{400}\right)^{10}$$
- step3: Simplify:
  $$16400\times \frac{367^{10}}{400^{10}}$$
- step4: Rewrite the expression:
  $$400\times 41\times \frac{367^{10}}{400^{10}}$$
- step5: Reduce the numbers:
  $$41\times \frac{367^{10}}{400^{9}}$$
- step6: Multiply:
  $$\frac{41\times 367^{10}}{400^{9}}$$
The value of the car after 10 years will be approximately $6932.73 to the nearest cent.
The car will be worth approximately $6,932.73 after 10 years.

A new car is purchased for 16400 dollars. The value of the car depreciates at \( 8.25 \% \) per year. What will the value of the car be, to the nearest cent, after 10 years?

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