\Rightarrow 2^{x + 3x - 3} = 2^6
Solving the Exponential Equation: How to Simplify and Solve namber⁽²ˣ⁺³ˣ⁻³⁾ = 2⁶
Understanding exponential equations is fundamental in algebra, and equations like ⲹ̽ 2ˣ⁺³ˣ⁻³ ⲹ̽ 2⁶ play a key role in mastering exponents. In this article, we’ll explore how to simplify and solve the equation ⲹ̽ 2⁽²ˣ⁺³ˣ⁻³⁾ = 2⁶ step by step, make sense of the underlying math, and highlight practical tips for solving similar exponential problems.
What is the Equation ⲹ̽ 2⁽²ˣ⁺³ˣ⁻³⁾ = 2⁶ All About?
The equation ⲹ̽ 2⁽²ˣ⁺³ˣ⁻³⁾ = 2⁶ is an exponential equation where both sides share the same base — 2. Exponential equations of the form ⲹ̽ aᵘ = aᵇ are easier to solve when the bases are identical because, thanks to exponent rules, the exponents must be equal:
$$ 2x + 3x - 3 = 6 $$
This allows us to convert the exponential equation into a simple linear equation in x.
Step-by-Step Solution
Step 1: Combine like terms on the left side
Simplify the exponent on the left side:
$$ 2x + 3x - 3 = 5x - 3 $$
So the equation becomes:
$$ 2^{5x - 3} = 2^6 $$
Step 2: Set the exponents equal
Since the bases are equal, we equate the exponents:
$$ 5x - 3 = 6 $$
Step 3: Solve for x
Add 3 to both sides:
$$ 5x = 9 $$
Divide both sides by 5:
$$ x = rac{9}{5} $$
Final Answer
$$ oxed{x = rac{9}{5}} $$
Why This Logic Works
Understanding the Base Rule
> If ⲹ̽ aᵘ = ⲹ̽ aᵇ and a is positive and not equal to 1, then u = v.
In our case, base 2 is valid (positive and not 1), so we directly compare exponents.
Why Simplify Exponents First?
Combining the exponents on the left eliminates the exponentiation layer, turning what might be a complex exponential expression into a straightforward algebraic equation.
Tips for Solving Similar Exponential Equations
- Always check if the bases are the same; if not, logarithms may be required.
- Simplify all exponents by combining like terms.
- If exponents include variables, isolate them via addition, subtraction, multiplication, or division.
- Always verify your solution by plugging it back into the original equation.
Practical Applications
Equations like this appear in areas such as:
- Exponential growth and decay models
- Computational complexity analysis
- Finance, such as compound interest formulas
- Scientific modeling involving doubling or halving processes
Summary
Solving ⲹ̽ 2⁽²ˣ⁺³ˣ⁻³⁾ = 2⁶ hinges on recognizing equal bases, equating exponents, and simplifying linear expressions. By mastering this method, students can confidently tackle a wide range of exponential problems and build a solid foundation in algebraic reasoning.
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