The amount in dollars in an interest-bearing account after t years is given by 500(1 + 0.02)4t. Which of these are equivalent to this expression? Select all that apply.

A.
500(1.061208)4t3


B.
(500⋅1.02)4t


C.
500(1.022⋅1.0404)t


D.
500(1.02)4t


E.
500(1.0404)2t


F.
(500⋅1.0404)2t



The given expression is \( 500(1 + 0.02)^{4t} \), which simplifies to \( 500(1.02)^{4t} \).

Now, let's go through each option to see which ones are equivalent to this:

**A. \( 500(1.061208)^{4t/3} \)**  
This is not equivalent. The base \( 1.061208 \) is an approximation of \( (1.02)^4 \), but the exponent \( \frac{4t}{3} \) does not match the original exponent \( 4t \). Therefore, **A is incorrect**.

**B. \( (500 \cdot 1.02)^{4t} \)**  
This is not equivalent either. The factor \( 500 \cdot 1.02 \) inside the parentheses changes the principal amount, so this expression does not match the original. **B is incorrect**.

**C. \( 500(1.02^2 \cdot 1.0404)^t \)**  
This expression can be rewritten as \( 500(1.0404)^t \cdot (1.02^2)^t \). Since this is not equivalent to the original base and exponent, **C is incorrect**.

**D. \( 500(1.02)^{4t} \)**  
This exactly matches the original expression \( 500(1.02)^{4t} \), so **D is correct**.

**E. \( 500(1.0404)^{2t} \)**  
Since \( 1.0404 \approx (1.02)^2 \), this is equivalent to \( 500((1.02)^2)^{2t} = 500(1.02)^{4t} \). Thus, **E is correct**.

**F. \( (500 \cdot 1.0404)^{2t} \)**  
This modifies the principal amount, similar to option B, which changes the meaning of the expression. So, **F is incorrect**.

### Correct options: **D and E**.

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Would you like more details on any part of this explanation?

Here are five related questions for further practice:
1. How do you simplify expressions with exponents in compound interest formulas?
2. What does the base of the exponential function represent in interest formulas?
3. How do changes in the base affect the growth rate in compound interest models?
4. How do you calculate continuous compound interest?
5. What is the difference between simple and compound interest?

**Tip:** Always check both the base and the exponent when determining if two exponential expressions are equivalent.

The amount in dollars in an interest-bearing account after t years is given by 500(1 + 0.02)4t. Which of these are equivalent to this expression? Select all that apply.

Explore the solution to a compound interest problem where you must identify equivalent exponential expressions for 500(1 + 0.02)4t. This guide covers key algebraic concepts such as exponential growth and compound interest formulas. Suitable for high school students in Grades 10-12.

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