## Podcast Beta

## Questions and Answers

Simplify $\frac{15^{2}e^{2}}{e^{2}}$.

$15^{2}$

Solve for $x$: $5^{x+3} imes 5^{x} = 5^{4}$.

$x = -2$

What is the estimated population of the village in 1990, given it was 753 in 1980 with a specific growth model?

Approximately 906

Solve for $K$: $(1+K)^{0.5} = 2$.

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Solve for $x$: $log_{10}(x + 2) = 2.5$.

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## Study Notes

### Simplifying Expressions

- Simplify expressions by rewriting them with positive indices only
- Examples of expressions to simplify:
- 2^(3/2-4)
- Q^(1.5)P
- √(8/(3^13+4))
- (15^2e^2)/e^2
- (3L^2Q^2/e^2)^2
- (3e^(0.1)/e)^2

### Solving Equations

- Solve exponential equations:
- 2^x = 1/16
- 5^(x+3)5^x = 5^4
- 2^p3^Q = 1/3
- (1+K)^(0.5) = 2
- e^(2x+2)/e^2 = 2

- Solve logarithmic equations:
- log10(x + 2) = 2.5
- 2ln(x) - ln(x + 1) = 0
- 2^(2x+1) = 7
- e^(x+1) = 1.5
- 20 + (2.4)^x^2 = 32.5

### Population Growth

- Population growth equation: P =
- Estimate population in:
- 1990
- 2000

- Find the year when the population reaches 1750 persons

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## Description

Simplify exponential expressions and solve equations involving exponents and logarithms, including solving for variables and evaluating numerical expressions.