\Rightarrow 2^x + 3x - 3 = 2^6 - Decision Point
Solving the Exponential Equation: How to Simplify and Solve namber⁽²ˣ⁺³ˣ⁻³⁾ = 2⁶
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.
Understanding the Context
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.
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Key Insights
Step-by-Step Solution
Step 1: Combine like terms on the left side
Simplify the exponent on the left side:
$$
2x + 3x - 3 = 5x - 3
$$
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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:
$$
x = rac{9}{5}
$$
