At the time of her grandson's birth, a grandmother deposits $\$ 5000$ in an account that pays $2 \%$ compounded monthly. What will be the value of the account at the child's twenty-first birthday, assuming that no other deposits or withdrawals are made during this period?
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The value of the account will be $\$$ (Round to the nearest dollar as needed.)

Step 1 ：The problem is asking for the future value of an investment with compound interest. The formula for compound interest is: \(FV = P * (1 + r/n)^{nt}\) where: \(FV\) is the future value, \(P\) is the principal amount (initial investment), \(r\) 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 ：In this case, \(P = \$5000\), \(r = 2\%\) or \(0.02\), \(n = 12\) (since it's compounded monthly), and \(t = 21\) years.

Step 3 ：We can plug these values into the formula to find the future value of the account: \(FV = 5000 * (1 + 0.02/12)^{12*21}\)

Step 4 ：Calculating the above expression, we get \(FV = 7607.147766058936\)

Step 5 ：Rounding to the nearest dollar, we get \(FV = 7607\)

Step 6 ：Final Answer: The value of the account at the child's twenty-first birthday will be \(\boxed{7607}\) dollars.