Alejandro Invests Money Into A College Savings Account. He Writes The Expression $750\left(1.025^{12}\right)^3$ To Help Him Calculate What The Account Balance Will Be In 3 Years. Write An Equivalent Expression That Could Represent Alejandro's

Introduction

Alejandro is planning for his future by investing money into a college savings account. To calculate the account balance in 3 years, he has written an expression $750\left(1.025^{12}\right)^3$. This expression represents the growth of his investment over time, taking into account the interest rate and the compounding frequency. In this article, we will explore the equivalent expression that could represent Alejandro's situation.

The Original Expression

The original expression is $750\left(1.025^{12}\right)^3$. This expression can be broken down into three main components:

• $750$: This is the initial investment amount.
• $1.025^{12}$: This represents the growth factor for one year, where $1.025$ is the interest rate and $12$ is the number of times the interest is compounded in a year.
• $^3$: This exponent indicates that the growth factor is raised to the power of $3$, representing the compounding of interest over three years.

Simplifying the Expression

To simplify the expression, we can use the property of exponents that states $(a^m)^n = a^{mn}$. Applying this property to the original expression, we get:

$$750\left(1.025^{12}\right)^3 = 750 \cdot 1.025^{36}$$

This simplified expression represents the same growth of the investment over three years, but with a more compact notation.

An Equivalent Expression

An equivalent expression that could represent Alejandro's situation is:

$$750 \cdot \left(1.025^3\right)^{12}$$

This expression is equivalent to the original expression because it represents the same growth of the investment over three years, but with a different order of operations. The exponent $3$ is raised to the power of $12$, representing the compounding of interest over three years, and then multiplied by the initial investment amount.

Why the Equivalent Expression is Useful

The equivalent expression $750 \cdot \left(1.025^3\right)^{12}$ is useful because it highlights the importance of the interest rate and the compounding frequency in determining the growth of the investment. By separating the interest rate and the compounding frequency, we can see that the interest rate is raised to the power of $3$, representing the compounding of interest over three years, and then multiplied by the initial investment amount.

Conclusion

In conclusion, the equivalent expression $750 \cdot \left(1.025^3\right)^{12}$ represents Alejandro's situation in a different way, but with the same underlying mathematics. This expression highlights the importance of the interest rate and the compounding frequency in determining the growth of the investment. By understanding the equivalent expression, we can gain a deeper insight into the mathematics behind Alejandro's college savings account.

Key Takeaways

• The original expression $750\left(1.025^{12}\right)^3$ represents the growth of Alejandro's investment over three years.
• The simplified expression $750 \cdot 1.025^{36}$ is equivalent to the original expression.
• The equivalent expression $750 \cdot \left(1.025^3\right)^{12}$ represents Alejandro's situation in a different way, but with the same underlying mathematics.
• The interest rate and the compounding frequency are crucial in determining the growth of the investment.

Further Reading

For more information on the mathematics behind college savings accounts, we recommend checking out the following resources:

• College Savings Plans: A comprehensive guide to college savings plans, including 529 plans and Coverdell ESAs.
• Compound Interest Calculator: A calculator that helps you calculate the future value of an investment based on the interest rate and the compounding frequency.
• Mathematics of Compound Interest: A video tutorial that explains the mathematics behind compound interest.
  Q&A: Understanding Alejandro's College Savings Account Expression ================================================================

Introduction

In our previous article, we explored the equivalent expression $750 \cdot \left(1.025^3\right)^{12}$ that represents Alejandro's college savings account. In this article, we will answer some frequently asked questions about the expression and provide additional insights into the mathematics behind college savings accounts.

Q: What is the interest rate in the expression?

A: The interest rate in the expression is 2.5% per year, which is represented by the value 1.025. This means that the investment will grow by 2.5% every year.

Q: How many times is the interest compounded in a year?

A: The interest is compounded 12 times in a year, which is represented by the exponent 12 in the expression. This means that the interest is applied 12 times in a year, resulting in a total growth of 2.5% per year.

Q: What is the initial investment amount?

A: The initial investment amount is $750, which is represented by the value 750 in the expression.

Q: How long does the investment grow for?

A: The investment grows for 3 years, which is represented by the exponent 3 in the expression.

Q: What is the equivalent expression for the investment growth?

A: The equivalent expression for the investment growth is $750 \cdot \left(1.025^3\right)^{12}$, which represents the same growth of the investment over 3 years.

Q: How can I calculate the future value of the investment?

A: To calculate the future value of the investment, you can use a compound interest calculator or the formula:

FV = PV x (1 + r)^n

Where: FV = Future Value PV = Present Value (initial investment amount) r = interest rate n = number of times the interest is compounded in a year

Q: What is the difference between the original expression and the equivalent expression?

A: The original expression $750\left(1.025^{12}\right)^3$ and the equivalent expression $750 \cdot \left(1.025^3\right)^{12}$ represent the same growth of the investment over 3 years, but with a different order of operations. The equivalent expression highlights the importance of the interest rate and the compounding frequency in determining the growth of the investment.

Q: Can I use the equivalent expression to calculate the investment growth for a different time period?

A: Yes, you can use the equivalent expression to calculate the investment growth for a different time period by changing the exponent 3 to the desired time period.

Conclusion

In conclusion, the equivalent expression $750 \cdot \left(1.025^3\right)^{12}$ represents Alejandro's college savings account in a different way, but with the same underlying mathematics. By understanding the equivalent expression, we can gain a deeper insight into the mathematics behind college savings accounts and calculate the future value of the investment.

Key Takeaways

• The interest rate in the expression is 2.5% per year.
• The interest is compounded 12 times in a year.
• The initial investment amount is $750.
• The investment grows for 3 years.
• The equivalent expression $750 \cdot \left(1.025^3\right)^{12}$ represents the same growth of the investment over 3 years.
• The formula FV = PV x (1 + r)^n can be used to calculate the future value of the investment.

Further Reading

For more information on the mathematics behind college savings accounts, we recommend checking out the following resources:

• College Savings Plans: A comprehensive guide to college savings plans, including 529 plans and Coverdell ESAs.
• Compound Interest Calculator: A calculator that helps you calculate the future value of an investment based on the interest rate and the compounding frequency.
• Mathematics of Compound Interest: A video tutorial that explains the mathematics behind compound interest.