Understanding 2^x · (2³)^{x−1} = 2⁶: Solving the Exponential Equation

When faced with the equation 2^x · (2³)^{x−1} = 2⁶, many students and learners wonder how to simplify and solve it efficiently. This problem beautifully demonstrates key principles of exponential expressions, especially the rules of exponents—leveraging powers of powers and product rules—making it a perfect example for practicing algebraic and logarithmic thinking in everyday math and education.

Understanding the Context

What does 2^x · (2³)^{x−1} = 2⁶ mean?

At first glance, this equation involves exponential terms with the same base—2—so simplifying it comes down to applying essential exponent rules:

  • Power of a power: (a^m)^n = a^{m·n}
  • Product of powers: a^m · a^n = a^{m+n}
2^x \cdot (2^3)^{x-1} = 2^6

Key Insights

Step-by-Step Simplification

Start with the original equation:

2^x · (2³)^{x−1} = 2⁶

Use the power of a power rule inside the parentheses:

2^x · [2^{3·(x−1)}] = 2⁶

Continue Reading

Set \( f'(x) = 0 \) to find critical points: \( 3x(x - 2) = 0 \).
So, \( x = 0 \) or \( x = 2 \).
Use the second derivative test: \( f''(x) = 6x - 6 \).

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Final Thoughts

Now apply the product rule:

2^{x + 3(x−1)} = 2⁶

Simplify the exponent on the left:

x + 3(x − 1) = x + 3x − 3 = 4x − 3

So the equation becomes:

2^{4x−3} = 2⁶

Since the bases are equal, set the exponents equal:

4x − 3 = 6

Solve for x:

4x = 6 + 3 = 9
x = 9/4