%5Ex-5%2F6.jpeg "(1 1/5)^x 5/6")
Exploring the Expression (1 1/5)^x * 5/6
This article delves into the intricacies of the expression (1 1/5)^x * 5/6, exploring its components, potential applications, and how to manipulate it.
Understanding the Components
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(1 1/5)^x: This portion represents a mixed number raised to a power, 'x'. The mixed number, 1 1/5, can be converted into an improper fraction, 6/5, for easier manipulation. The exponent 'x' signifies repeated multiplication of the base (6/5) by itself.
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5/6: This is a simple fraction, representing a part of a whole. It acts as a multiplier to the result obtained from the first component.
Applications and Interpretations
The expression can find applications in various fields:
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Exponential Growth/Decay: If 'x' represents time, the expression could model scenarios involving exponential growth or decay. For instance, imagine a population of bacteria doubling every hour, with the initial population being 6/5 of a unit. The expression would represent the population at any given time 'x'.
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Geometric Sequences: The expression can be used to represent the terms of a geometric sequence, where the first term is 5/6 and the common ratio is 6/5.
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Compound Interest: The expression can be used to calculate the amount of money accumulated over time when interest is compounded.
Manipulating the Expression
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Simplifying the Base: As mentioned earlier, (1 1/5) can be converted to 6/5. This simplifies the expression to (6/5)^x * 5/6.
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Exponent Rules: The power of a fraction can be distributed to the numerator and denominator, resulting in (6^x/5^x) * 5/6.
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Simplifying Multiplication: The 5 in the numerator of the fraction and the 5 in the denominator of the power can be canceled, leaving (6^x / 5^(x-1)) * 1/6.
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Combining Terms: The 1/6 can be brought inside the fraction to get (6^(x-1) / 5^(x-1)).
Conclusion
Understanding the components and potential applications of (1 1/5)^x * 5/6 is crucial in various mathematical and scientific contexts. By manipulating the expression, we can gain deeper insights into its behavior and use it effectively in real-world scenarios.