Luke and Karen deposit $$ \$ 700.00 $$ into a savings account which earns $$ 14 \% $$ interest compounded monthly. They want to use the money in the account to $$ g 0 $$ on a trip in 3 years. How much will they be able to spend? Use the formula $$ A=P\left(1+\frac{r}{n}\right)^{n t} $$, where $$ A $$ is the balance (final amount), $$ P $$ is the principal (starting amount), $$ r $$ is the interest rate expressed as a decimal, $$ n $$ is the number of times per year that the interest is compounded, and $$ t $$ is the time in years. Round your answer to the nearest cent.

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To find out how much Luke and Karen will be able to spend on their trip in 3 years, we need to calculate the balance in their savings account after 3 years with the given interest rate and compounding frequency.

Given:
- Principal (P) = $700.00
- Interest Rate (r) = 14% = 0.14 (as a decimal)
- Compounding Frequency (n) = 12 (monthly)
- Time (t) = 3 years

We can use the formula for compound interest:
$$ A = P\left(1 + \frac{r}{n}\right)^{nt} $$

Substitute the given values into the formula:
$$ A = 700.00\left(1 + \frac{0.14}{12}\right)^{12 \times 3} $$

Now, we can calculate the balance in the savings account after 3 years.
Calculate the value by following steps:
- step0: Calculate:
  $$700\left(1+\frac{0.14}{12}\right)^{12\times 3}$$
- step1: Divide the terms:
  $$700\left(1+\frac{7}{600}\right)^{12\times 3}$$
- step2: Add the numbers:
  $$700\left(\frac{607}{600}\right)^{12\times 3}$$
- step3: Multiply the numbers:
  $$700\left(\frac{607}{600}\right)^{36}$$
- step4: Simplify:
  $$700\times \frac{607^{36}}{600^{36}}$$
- step5: Rewrite the expression:
  $$100\times 7\times \frac{607^{36}}{600^{36}}$$
- step6: Rewrite the expression:
  $$100\times 7\times \frac{607^{36}}{100^{36}\times 6^{36}}$$
- step7: Reduce the numbers:
  $$7\times \frac{607^{36}}{100^{35}\times 6^{36}}$$
- step8: Multiply the fractions:
  $$\frac{7\times 607^{36}}{100^{35}\times 6^{36}}$$
The balance in the savings account after 3 years will be approximately $1062.79.

Therefore, Luke and Karen will be able to spend $1062.79 on their trip in 3 years.
Luke and Karen will have approximately $1,062.79 in their savings account after 3 years, so they can spend about $1,062.79 on their trip.

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