Exploring Exponential Growth: Why 75 × (1.12)³ Equals 105.3696

In the world of math, exponents aren’t just abstract symbols—they model powerful real-world phenomena like growth, investment, and population dynamics. One practical example is calculating compound growth: how an initial value increases over time with consistent percentages. A classic calculation that demonstrates this principle is:

75 × (1.12)³ = 105.3696

Understanding the Context

This equation reflects exponential growth and helps us understand how steady percentage increases compound over time. Let’s unpack it step-by-step and explore its significance.

What Does 75 × (1.12)³ Represent?

At its core, this expression models growth scenarios where something increases by 12% each period. For instance:

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Key Insights

  • Finance: An investment of $75 that grows at 12% annually for three years.
  • Population: A community growing at 12% per year over three years.
  • Science and Industry: A microbial culture or chemical reaction multiplying by 12% each hour or day.

In each context, the growth compounds—meaning each period’s increase is calculated on the new, higher value—not just the original amount.

Breaking Down the Calculation

Let’s compute how the equation unfolds:

  1. Base value: Start with 75
  2. Growth factor: The annual increase is 12%, which as a decimal is 1.12
  3. Time period: This growth applies over 3 periods (e.g., years)

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Each gives a perfect square divisible by 4: $4, 16, 36, 64, 100, 144, 196$.
Solution: Consider any four consecutive integers: $n, n+1, n+2, n+3$.
Solution: Let the side length of the square be $ s $. The diagonal of a square divides it into two congruent right triangles, and the relationship between the side length and the diagonal is given by the Pythagorean theorem:

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Final Thoughts

Now plug into the formula:
75 × (1.12)³

First, calculate (1.12)³:
1.12 × 1.12 = 1.2544
Then, 1.2544 × 1.12 = 1.404928

Now multiply:
75 × 1.404928 = 105.3696

Why 105.3696?

The result, 105.3696, shows the total after three consecutive 12% increases. This demonstrates compounding effect—small, consistent growth accumulates significantly over time.

For example:

  • After year 1: 75 × 1.12 = 84
  • After year 2: 84 × 1.12 = 94.08
  • After year 3: 94.08 × 1.12 = 105.3696

This method highlights the power of exponential growth—something familiar in saving money, investing in stocks, or even modeling natural population increases.

Real-World Applications of Exponential Growth

Understanding such calculations helps in: