Which of the following is exponential growth?

To identify exponential growth and decay and simplify the expressions, let's analyze each option.

Exponential Growth

Exponential growth takes the form \(y = a(1 + r)^t\), where \(r\) is a positive rate.

1. \(y = 200(1 - 0.12)^t\) - This is not exponential growth; it's a decay scenario.
2. \(y = 500(0.98)^{3t}\) - This represents exponential decay, since \(0.98 < 1\).
3. \(y = 750(1 - 0.45)^t\) - This is also not exponential growth; it's decay.
4. \(y = 1,000(1 + 0.27)^{4t}\) - This represents exponential growth because \(1 + 0.27 = 1.27 > 1\).

Thus, the answer for exponential growth is: y = 1,000(1 + 0.27)^{4t}

Exponential Decay of 55%

To show exponential decay of 55%, we can express it as: \(y = a(1 - r)^t\) where \(r = 0.55\), meaning \(1 - r = 0.45\).

1. \(y = 200(1 - 0.12)^t\) - Not decay of 55%.
2. \(y = 500(0.98)^{3t}\) - Not decay of 55%.
3. \(y = 750(1 - 0.45)^t\) - This indicates decay of 45%, not 55%.
4. \(y = 1,000(1 + 0.27)^{4t}\) - Not decay; it's growth.

In fact, none of these options represent decay of 55% directly, however, with the mentioned consideration of \(r = 0.55\), the ideal format would be \(y = a(1 - 0.55)^t\). Thus, the answer is: None of the provided options show 55% decay.

Simplification

To simplify an expression, I would need the specific mathematical expression to work with. If simplifying a fraction, for instance: \[ \frac{8}{4} \] This simplifies to: \[ 2 \]

Solve the following

Without a specified equation or expression, I cannot provide an answer. If provided, the numerical calculation will be performed accordingly.

Please provide explicit mathematical expressions for simplification or solving that you would like assistance with!