× (1.12)^3 = 75 × 1.404928 = 105.3696.
Solving × (1.12)³ = 75 × 1.404928 = 105.3696: A Detailed Breakdown
Understanding how to solve exponential equations and their real-world applications is essential in mathematics, science, and various technical fields. Today, we explore the equation:
> × (1.12)³ = 75 × 1.404928 = 105.3696
We’ll break down each component step-by-step, explain the math behind it, and clarify how this expression reveals key numeric relationships—ultimately confirming the equality and discussing its practical significance.
What Does the Equation Mean?
The main equation is: × (1.12)³ = 75 × 1.404928 = 105.3696
Our goal is to solve for the unknown multiplier (×) while verifying the entire expression step-by-step.
Step 1: Simplify the Right-Hand Side
Start with the right side of the equation:
> 75 × 1.404928 = 105.3696
Why this matters: Confirming this multiplication validates we’re working with accurate constants, a necessary foundation before isolating the variable.
Calculation: 75 × 1.404928 = 105.3696
✅ The right side checks out.
Step 2: Isolate × Using Exponential Properties
Given: × (1.12)³ = 105.3696
To solve for ×, divide both sides by (1.12)³:
> × = 105.3696 / (1.12)³
Step 3: Compute (1.12)³
Calculate the cube:
(1.12)³ = 1.12 × 1.12 × 1.12
🔹 First: 1.12 × 1.12 = 1.2544 🔹 Then: 1.2544 × 1.12 ≈ 1.404928
This confirms the given value 1.404928, validating the use of this constant.
Step 4: Final Division
Now divide:
105.3696 / 1.404928 ≈ 75
Using precise calculation: 105.3696 ÷ 1.404928 = 75
✅ This confirms the coefficient on the left matches perfectly: the unknown × equals 75.
Full Solution
Putting it all together: × = 75 and (1.12)³ ≈ 1.404928, so:
75 × (1.12)³ ≈ 105.3696
Practical Applications & Why It Matters
Equations involving exponential growth or decay—such as compound interest, population growth, or radioactive decay—are often modeled with expressions like A × (1 + r)ⁿ, where:
- A = principal amount or initial value
- r = growth or decay rate
- n = number of time periods
In our case:
- A = 75
- r = 12% = 0.12
- n = 3 years (cubed)
- Constant factor = ~1.404928, representing compounded growth
Thus, × (1.12)³ = 75 × 1.404928 = 105.3696 models a 12% annual increase over three years, climbing from 75 to approximately 105.37.
Key Takeaways
- Precise calculation ensures accuracy when dealing with exponents.
- Breaking down expressions reveals their logical structure and factual basis.
- Exponential models like (1.12)³ reflect real-world growth dynamics.
- Verifying constants like 1.404928 improves problem-solving confidence.
Conclusion
The equation × (1.12)³ = 75 × 1.404928 = 105.3696 elegantly demonstrates the application of exponents and algebraic manipulation. By systematically isolating the unknown and validating constants, we confirm this expression accurately represents compounded growth over three periods.
Whether modeling economics, biology, or technology, mastering such equations empowers you to analyze and predict dynamic systems confidently.
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