Rounded to the nearest cent, find the amount after 2 years if $\$ 400$ is invested at $7 \%$ compounded:
a. Annually

Amount $=\$$
b. Quarterly

Amount $=\$$
c. Continuously

Amount $=\$$
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Step 1 ：The formula for compound interest is \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years.

Step 2 ：For annually, \(n = 1\). So, the formula becomes \(A = P(1 + r)^{t}\).

Step 3 ：Substituting the given values, we get \(A = 400(1 + 0.07/1)^{1*2}\), which simplifies to \(A = 400(1.07)^2\), then to \(A = 400 * 1.1449\), and finally to \(A = 457.96\).

Step 4 ：\(\boxed{457.96}\) is the amount after 2 years if compounded annually.

Step 5 ：For quarterly, \(n = 4\). So, the formula becomes \(A = P(1 + \frac{r}{n})^{nt}\).

Step 6 ：Substituting the given values, we get \(A = 400(1 + 0.07/4)^{4*2}\), which simplifies to \(A = 400(1.0175)^8\), then to \(A = 400 * 1.150781\), and finally to \(A = 460.31\).

Step 7 ：\(\boxed{460.31}\) is the amount after 2 years if compounded quarterly.

Step 8 ：The formula for continuously compounded interest is \(A = Pe^{rt}\), where \(e\) is the base of the natural logarithm (approximately equal to 2.71828).

Step 9 ：Substituting the given values, we get \(A = 400e^{0.07*2}\), which simplifies to \(A = 400 * e^{0.14}\), then to \(A = 400 * 1.150273\), and finally to \(A = 460.11\).

Step 10 ：\(\boxed{460.11}\) is the amount after 2 years if compounded continuously.