A new car is purchased for 16200 dollars. The value of the car depreciates at $$ 14.25 \% $$ per year. What will the value of the car be, to the nearest cent, after 6 years?

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To find the value of the car after 6 years, we can use the formula for depreciation:

$$ \text{Value after depreciation} = \text{Initial value} \times (1 - \text{Depreciation rate})^{\text{Number of years}} $$

Given:
- Initial value of the car = $16200
- Depreciation rate = 14.25% per year
- Number of years = 6 years

Substitute the values into the formula:

$$ \text{Value after depreciation} = 16200 \times (1 - 0.1425)^6 $$

Now, we can calculate the value of the car after 6 years.
Calculate the value by following steps:
- step0: Calculate:
  $$16200\left(1-0.1425\right)^{6}$$
- step1: Subtract the numbers:
  $$16200\times 0.8575^{6}$$
- step2: Convert the expressions:
  $$16200\left(\frac{343}{400}\right)^{6}$$
- step3: Simplify:
  $$16200\times \frac{343^{6}}{400^{6}}$$
- step4: Rewrite the expression:
  $$200\times 81\times \frac{343^{6}}{400^{6}}$$
- step5: Rewrite the expression:
  $$200\times 81\times \frac{343^{6}}{200^{6}\times 2^{6}}$$
- step6: Reduce the numbers:
  $$81\times \frac{343^{6}}{200^{5}\times 2^{6}}$$
- step7: Multiply the fractions:
  $$\frac{81\times 343^{6}}{64\times 200^{5}}$$
The value of the car after 6 years will be approximately $6440.50 to the nearest cent.
The car's value after 6 years is approximately $6,440.50.

A new car is purchased for 16200 dollars. The value of the car depreciates at \( 14.25 \% \) per year. What will the value of the car be, to the nearest cent, after 6 years?

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